**Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables**

Gabino Torres-Vega

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53598

### **1. Introduction**

There are many proposals for writing Classical and Quantum Mechanics in the same lan‐ guage. Some approaches use complex functions for classical probability densities [1] and other define functions of two variables from single variable quantum wave functions [2,3]. Our approach is to use the same concepts in both types of dynamics but in their own realms, not using foreign unnatural objects. In this chapter, we derive many inter relationships be‐ tween conjugate variables.

#### **1.1. Conjugate variables**

An important object in Quantum Mechanics is the eigenfunctions set {|*n* >} *n*=0 *∞* of a Hermi‐ tian operator *F* ^ . These eigenfunctions belong to a Hilbert space and can have several repre‐ sentations, like the coordinate representation *ψn*(*q*)= *q* | *n* . The basis vector used to provide the coordinate representation, |*q* >, of the wave function are themselves eigenfunctions of the coordinate operator *Q* ^ We proceed to define the classical analogue of both objects, the eigenfunction and its support.

Classical motion takes place on the associated cotangent space *T \* Q* with variable *z* =(*q*, *p*), where *q* and *p* are *n* dimensional vectors representing the coordinate and momentum of point particles. We can associate to a dynamical variable *F* (*z*) its eigensurface, i.e. the level set

$$\Sigma\_F(f) = \{ z \in T \: \mathbb{Q} \mid F(z) = f \} \tag{1}$$

© 2013 Torres-Vega; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Torres-Vega; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Where *f* is a constant, one of the values that *F* (*z*) can take. This is the set of points in phase space such that when we evaluate *F* (*z*), we obtain the value *f* . Examples of these eigensurfa‐ ces are the constant coordinate surface, *q* = *X* , and the energy shell, *H* (*z*)=*E*, the surface on which the evolution of classical systems take place. These level sets are the classical analogues of the support of quantum eigenfunctions in coordinate or momentum representations.

Many dynamical variables come in pairs. These pairs of dynamical variables are related through the Poisson bracket. For a pair of conjugate variables, the Poisson bracket is equal to one. This is the case for coordinate and momentum variables, as well as for energy and time. In fact, according to Hamilton's equations of motion, and the chain rule, we have that

$$\{\{t\_{\prime}\}\}=\sum\_{i}\left(\frac{\partial \, t}{\partial \, q^{i}}\frac{\partial \, H}{\partial \, p\_{i}}-\frac{\partial \, H}{\partial \, q^{i}}\frac{\partial \, t}{\partial \, p\_{i}}\right) = \sum\_{i}\left(\frac{dt}{dq^{i}}\frac{d \, q^{i}}{dt}+\frac{d \, \, p\_{i}}{dt}\frac{dt}{dq\_{i}}\right) = \frac{dt}{dt} = \mathbf{1} \tag{2}$$

A similar situation is found with the conjugate pair energy-time. Usually the energy is well defined in phase space but time is not. In a previous work, we have developed a method for defining a time coordinate in phase space [4]. The method takes the hypersurface *q* 1 = *X* , where *X* is fixed, as the zero time eigensurface and propagates it forward and backward in

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

Now, recall that any phase space function *G*(*z*) generates a motion in phase space through a

where *f* is a variable with the same units as the conjugate variable *F* (*z*). You can think of *G*(*z*) as the Hamiltonian for a mechanical system and that *f* is the time. For classical sys‐ tems, we are considering conjugate pairs leading to conjugate motions associated to each variable with the conjugate variable serving as the evolution parameter (see below). This will be applied to the energy-time conjugate pair. Let us derive some properties in which the

Some relationships between a pair of conjugate variables are derived in this section. We will deal with general *F* (*z*) and *G*(*z*) conjugate variables, but the results can be applied to coordi‐

is the classical analogue of the corresponding quantum eigenstate in coordinate *q* | *g* and momentum *p* | *g* representations. When *G*(*z*) is evaluated at the points of the support of (*z* | *g*), we get the value *g*. We use a bra-ket like notation to emphasise the similarity with

The magnitude of the vector field | *XG* | is the change of length along the *f* direction

∂ *p* , - ∂ *G*

∂ *q* ), (4)

http://dx.doi.org/10.5772/53598

5

*df* , (5)

(*z*|*g*)=*δ*(*z* - *v*), *v* ∈Σ*G*(*g*) (6)

^ or *G*

^ provides margin‐

time generating that way a coordinate system for time in phase space.

*df* = *XG*, *XG* =( ∂ *G*

nate and momentum or energy and time or to any other conjugate pair.

is the length element.

The overlap between a probability density with an eigenfunction of *F*

|*XG*|= *d q i df d q i df* + *d pi df d pi df* = *d lF*

A unit density with the eigensurface Σ*G*(*g*) as support

set of symplectic system of equations, a dynamical system,

*dz*

two conjugate variables participate.

where *dlF* = (*dq i*)2 + (*d p i*)2

the quantum concepts.

al representations of a probability density,

**1.3. The interplay between conjugate variables**

Now, a point in cotangent space can be specified as the intersection of 2*n* hypersurfaces. A set of 2*n* independent, intersecting, hypersurfaces can be seen as a coordinate system in co‐ tangent space, as is the case for the hyper surfaces obtained by fixing values of coordinate and momentum, i.e. the phase space coordinate system with an intersection at *z* =(*q*, *p*). We can think of alternative coordinate systems by considering another set of conjugate dynami‐ cal variables, as is the case of energy and time.

Thus, in general, the *T \* Q* points can be represented as the intersection of the eigensurfaces of the pair of conjugate variables *F* and *G*,

$$\Sigma\_{\rm FG}(f,\,\,\,\mathbf{g}) = \left\{ \mathbf{z} \in T \, \, \, \middle| \, \, \, \, \mathcal{F}(\mathbf{z}) = f, \,\, \, \mathcal{G}(\mathbf{z}) = \mathbf{g} \right\}.\tag{3}$$

A point in this set will be denoted as an abstract bra ( *f* , *g* |, such that ( *f* , *g* |*u*) means the function *u*( *f* , *g*).

We can also have marginal representations of functions in phase space by using the eigen‐ surfaces of only one of the functions,

$$\Sigma\_{\Gamma}(f) = \left\{ z \in T \: \bigotimes^\* F(z) = f \right\}, \quad \text{and} \quad \Sigma\_{G}(g) = \left\{ z \in T \: \bigotimes^\* Q \mid G(z) = g \right\}.$$

A point in the set Σ*F* ( *f* ) Σ*G*(*g*) will be denoted by the bra ( *f* | (*g* | and an object like ( *f* | *u*) (*g* |*u*) will mean the *f g* dependent function *u*( *f* ) *u*(*g*) .

#### **1.2. Conjugate coordinate systems**

It is usual that the origin of one of the variables of a pair of conjugate variables is not well defined. This happens, for instance, with the pair of conjugate variables *q* and *p*. Even though the momentum can be well defined, the origin of the coordinate is arbitrary on the trajectory of a point particle, and it can be different for each trajectory. A coordinate system fixes the origin of coordinates for all of the momentum eigensurfaces.

A similar situation is found with the conjugate pair energy-time. Usually the energy is well defined in phase space but time is not. In a previous work, we have developed a method for defining a time coordinate in phase space [4]. The method takes the hypersurface *q* 1 = *X* , where *X* is fixed, as the zero time eigensurface and propagates it forward and backward in time generating that way a coordinate system for time in phase space.

Now, recall that any phase space function *G*(*z*) generates a motion in phase space through a set of symplectic system of equations, a dynamical system,

$$\frac{dz}{d\bar{f}} = X\_{G'} \qquad \qquad X\_G = \begin{pmatrix} \frac{\partial G}{\partial p} \; \; \; \; \; \; \; \; \frac{\partial G}{\partial q} \; \; \; \; \; \; \tag{4}$$

where *f* is a variable with the same units as the conjugate variable *F* (*z*). You can think of *G*(*z*) as the Hamiltonian for a mechanical system and that *f* is the time. For classical sys‐ tems, we are considering conjugate pairs leading to conjugate motions associated to each variable with the conjugate variable serving as the evolution parameter (see below). This will be applied to the energy-time conjugate pair. Let us derive some properties in which the two conjugate variables participate.

#### **1.3. The interplay between conjugate variables**

Where *f* is a constant, one of the values that *F* (*z*) can take. This is the set of points in phase space such that when we evaluate *F* (*z*), we obtain the value *f* . Examples of these eigensurfa‐ ces are the constant coordinate surface, *q* = *X* , and the energy shell, *H* (*z*)=*E*, the surface on which the evolution of classical systems take place. These level sets are the classical analogues of the support of quantum eigenfunctions in coordinate or momentum representations.

Many dynamical variables come in pairs. These pairs of dynamical variables are related through the Poisson bracket. For a pair of conjugate variables, the Poisson bracket is equal to one. This is the case for coordinate and momentum variables, as well as for energy and time. In fact, according to Hamilton's equations of motion, and the chain rule, we have that

Now, a point in cotangent space can be specified as the intersection of 2*n* hypersurfaces. A set of 2*n* independent, intersecting, hypersurfaces can be seen as a coordinate system in co‐ tangent space, as is the case for the hyper surfaces obtained by fixing values of coordinate and momentum, i.e. the phase space coordinate system with an intersection at *z* =(*q*, *p*). We can think of alternative coordinate systems by considering another set of conjugate dynami‐

A point in this set will be denoted as an abstract bra ( *f* , *g* |, such that ( *f* , *g* |*u*) means the

We can also have marginal representations of functions in phase space by using the eigen‐

A point in the set Σ*F* ( *f* ) Σ*G*(*g*) will be denoted by the bra ( *f* | (*g* | and an object like

It is usual that the origin of one of the variables of a pair of conjugate variables is not well defined. This happens, for instance, with the pair of conjugate variables *q* and *p*. Even though the momentum can be well defined, the origin of the coordinate is arbitrary on the trajectory of a point particle, and it can be different for each trajectory. A coordinate system

*Q* points can be represented as the intersection of the eigensurfaces

*Q* | *G*(*z*)= *g*}.

*Q* | *F*(*z*)= *f* , *G*(*z*)= *g*}. (3)

*dt* =1 (2)

{*t*, *H* }=∑ *i* ( ∂ *t* ∂ *q i* ∂ *H* ∂ *pi* - ∂ *H* ∂ *q i* ∂ *t* ∂ *pi* ) =∑ *i* ( *dt d q i d q i dt* + *d pi dt dt d pi* ) = *dt*

cal variables, as is the case of energy and time.

Σ*FG*( *f* , *g*)={*z* ∈*T* \*

*Q* | *F* (*z*)= *f* } , and Σ*G*(*g*)={*z* ∈*T* \*

( *f* | *u*) (*g* |*u*) will mean the *f g* dependent function *u*( *f* ) *u*(*g*) .

fixes the origin of coordinates for all of the momentum eigensurfaces.

of the pair of conjugate variables *F* and *G*,

surfaces of only one of the functions,

**1.2. Conjugate coordinate systems**

Thus, in general, the *T \**

4 Advances in Quantum Mechanics

function *u*( *f* , *g*).

Σ*F* ( *f* )={*z* ∈*T* \*

Some relationships between a pair of conjugate variables are derived in this section. We will deal with general *F* (*z*) and *G*(*z*) conjugate variables, but the results can be applied to coordi‐ nate and momentum or energy and time or to any other conjugate pair.

The magnitude of the vector field | *XG* | is the change of length along the *f* direction

$$\mathbb{I}\mid \mathbf{X}\_{\leq \mathbf{f}}\mid = \sqrt{\frac{d\,q^i}{df}\,\frac{d\,q^i}{df} + \frac{d\,p\_i}{df}\frac{d\,p\_i}{df}} = \frac{d\,l\_F}{df}\,,\tag{5}$$

where *dlF* = (*dq i*)2 + (*d p i*)2 is the length element.

A unit density with the eigensurface Σ*G*(*g*) as support

$$(z \mid \mathbf{g}) = \delta(z \text{ - } v), \quad v \in \Sigma\_G(\mathbf{g}) \tag{6}$$

is the classical analogue of the corresponding quantum eigenstate in coordinate *q* | *g* and momentum *p* | *g* representations. When *G*(*z*) is evaluated at the points of the support of (*z* | *g*), we get the value *g*. We use a bra-ket like notation to emphasise the similarity with the quantum concepts.

The overlap between a probability density with an eigenfunction of *F* ^ or *G* ^ provides margin‐ al representations of a probability density,

$$\log(f) \coloneqq \{ f \mid \rho \} \coloneqq \mathfrak{f}(f \mid \mathbf{z}) \langle \mathbf{z} \mid \rho \rangle dz = \mathfrak{f} \delta(\mathbf{z} \cdot f) \rho(\mathbf{z}) dz \,, \quad f \in \sum\_{\mathbf{F}} \langle f \rangle. \tag{7}$$

$$\mathfrak{g}(\emptyset) \coloneqq \mathfrak{g} \mid \mathfrak{\rho} \rangle = \mathfrak{f}(\mathfrak{g} \mid \mathfrak{z})(\mathfrak{z} \mid \mathfrak{\rho})dz = \mathfrak{f}\mathfrak{f}(\mathfrak{z} \cdot \mathfrak{g})\mathfrak{\rho}(\mathfrak{z})dz \,, \quad \mathfrak{g} \in \sum\_{\mathbb{C}} \mathfrak{k}\mathfrak{g} \,. \tag{8}$$

Note that *F* is at the same time a parameter in terms of which the motion of points in phase

*dG* = - ∂ *F*

*dF* = *XG* , *XG* =( ∂ *G*

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

Now, *G* is the shift parameter besides of being the conjugate variable to *F*. This also renders

*d q i dG* ∂ *G* ∂ *q i* = *dG*

Then, the motion along one of the *F* or *G* directions is determined by the corresponding

If the motion of phase space points is governed by the vector field (15), *F* remains constant

In contrast, when motion occurs in the *F* direction, by means of Eq. (16), it is the *G* variable

∂ *pi* ∂ *F* ∂ *q i* ∂ *F* +

Hence, motion originated by the conjugate variables *F* (*z*) and *G*(*z*) occurs on the shells of

∂ *q i* =0 , ∇ ∙ *XG* = ∂

∂ *q i* ∂ *F* ∂ *pi*

> ∂ *q i* ∂ *G* ∂ *pi* - ∂ ∂ *pi* ∂ *G*

∂ *pi* . (13)

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7

*dG* =1 . (14)

, - ∂ *G*

∂ *G* =0 . (16)

∂ *F* =0 . (17)

∂ *q i* =0 . (18)

∂ *q i* ) (15)

∂ *pi*

∂ *q i* , *d q i*

space is written, and also the conjugate variable to *G*.

*d pi dG* = ∂ *F*

{*F* , *G*}= *d pi dG* ∂ *G* ∂ *pi* +

> ∂ *F* ∂ *pi* , ∂ *F*

The dynamical systems and vector fields for the motions just defined are

∂ *q i* ) , and *dz*

conjugate variable. These vector fields in general are not orthogonal, nor parallel.

We can also define other dynamical system as

the Poisson bracket to the identity

*dG* = *XF* , *XF* =(-

*dF dG* = ∂ *F* ∂ *q i* ∂ *q i* ∂ *G* + ∂ *F* ∂ *pi* ∂ *pi* ∂ *G* = ∂ *pi* ∂ *G* ∂ *qi* ∂ *G* - ∂ *q i* ∂ *G* ∂ *pi*

the one that remains constant because

*dG dF* = ∂ *G* ∂ *q i* ∂ *q i* ∂ *F* + ∂ *G* ∂ *pi* ∂ *pi* ∂ *F* = -

constant *F* (*z*) or of constant *G*(*z*), respectively.

The divergence of these vector fields is zero,

∂ *q i* ∂ *F* ∂ *pi* +

∂ ∂ *pi* ∂ *F*

∇ ∙ *XF* = - ∂

*dz*

because

But, a complete description of a function in *T \* Q* is obtained by using the two dimensions unit density (*z* | *f* , *g*)=*δ*(*z* - ( *f* , *g*)), the eigenfunction of a location in phase space,

$$\mathfrak{g}(f,\,\,\mathbf{g}) = \{f,\,\,\mathbf{g} \mid \rho\} = \mathfrak{f}(f,\,\,\mathbf{g} \mid \mathbf{z}) \langle \mathbf{z} \mid \rho \rangle dz = \mathfrak{f} \delta(z \,\,\mathbf{z} \,\, \mathbf{f} \,\, \mathbf{g}) \rho(\mathbf{z}) dz \,, \,, \quad \left(f,\,\,\mathbf{g}\right) \in \sum\_{\mathcal{C}} \langle f,\,\,\mathbf{g} \rangle. \tag{9}$$

In this way, we have the classical analogue of the quantum concepts of eigenfunctions of op‐ erators and the projection of vectors on them.

#### **1.4. Conjugate motions**

Two dynamical variables with a constant Poisson bracket between them induce two types of complementary motions in phase space. Let us consider two real functions *F*(*z*) and *G*(*z*) of points in cotangent space *z* ∈ *T\* Q* of a mechanical system, and a unit Poisson bracket be‐ tween them,

$$\{F, \,\,\,\mathbf{G}\} = \frac{\partial F}{\partial q^i}, \frac{\partial G}{\partial p\_i} - \frac{\partial G}{\partial q^i}, \frac{\partial F}{\partial p\_i} = \mathbf{1} \,\,\,\,\tag{10}$$

valid on some domain D=D( ∂ *F* ∂ *q i* ) ∩D( ∂ *F* ∂ *q i* ) ∩D( ∂ *F* ∂ *q i* ) ∩D( ∂ *F* ∂ *q i* ), according to the considered func‐ tions *F* and *G*. The application of the chain rule to functions of *p* and *q*, and Eq. (10), sug‐ gests two ways of defining dynamical systems for functions *F* and *G* that comply with the unit Poisson bracket. One of these dynamical systems is

$$\frac{d^i p\_i^i}{dF^i} = 1 \cdot \frac{\partial \, G}{\partial \, q^i} \,, \qquad \frac{d \, q^i}{dF^i} = \frac{\partial \, G}{\partial \, p\_i^i} \,, \tag{11}$$

With these replacements, the Poisson bracket becomes the derivative of a function with re‐ spect to itself

$$\{F\_{\nu}, G\} = \frac{\partial \, F}{\partial q^{i}} \frac{\partial q^{i}}{\partial F} + \frac{\partial \, p\_{i}}{\partial F} \frac{\partial F}{\partial p\_{i}} = \frac{\mathrm{d}F}{\mathrm{d}F} = \mathbf{1} \,\tag{12}$$

Note that *F* is at the same time a parameter in terms of which the motion of points in phase space is written, and also the conjugate variable to *G*.

We can also define other dynamical system as

ρ( *f* )≔( *f* | *ρ*)≔*∫*( *f* | *z*)(*z* | *ρ*)*dz* =*∫δ*(*z* - *f* )*ρ*(*z*)*dz* , *f* ∈∑

ρ(*g*)≔(*g* | *ρ*)≔*∫*(*g* | *z*)(*z* | *ρ*)*dz* =*∫δ*(*z* - *g*)*ρ*(*z*)*dz* , *g* ∈∑

unit density (*z* | *f* , *g*)=*δ*(*z* - ( *f* , *g*)), the eigenfunction of a location in phase space,

In this way, we have the classical analogue of the quantum concepts of eigenfunctions of op‐

Two dynamical variables with a constant Poisson bracket between them induce two types of complementary motions in phase space. Let us consider two real functions *F*(*z*) and *G*(*z*) of

ρ( *f* , g)≔( *f* , *g* | *ρ*)≔*∫*( *f* , *g* | *z*)(*z* | *ρ*)*dz* =*∫δ*(*z* - ( *f* , *g*))*ρ*(*z*)*dz* , ( *f* , *g*)∈ ∑

{*F* , *G*}= ∂ *F* ∂ *q i* ∂ *G* ∂ *pi* - ∂ *G* ∂ *q i* ∂ *F*

∂ *q i* ) ∩D( ∂ *F*

unit Poisson bracket. One of these dynamical systems is

*d pi dF* = - ∂ *G*

{*F* , *G*}= ∂ *F* ∂ *q i* ∂ *q i* ∂ *F* + ∂ *pi* ∂ *F* ∂ *F* ∂ *pi* = d*F*

∂ *q i* ) ∩D( ∂ *F*

∂ *q i* , *d q i*

With these replacements, the Poisson bracket becomes the derivative of a function with re‐

tions *F* and *G*. The application of the chain rule to functions of *p* and *q*, and Eq. (10), sug‐ gests two ways of defining dynamical systems for functions *F* and *G* that comply with the

∂ *q i* ) ∩D( ∂ *F*

*dF* = ∂ *G*

But, a complete description of a function in *T \**

erators and the projection of vectors on them.

**1.4. Conjugate motions**

6 Advances in Quantum Mechanics

tween them,

spect to itself

points in cotangent space *z* ∈ *T\**

valid on some domain D=D( ∂ *F*

*F*

*G*

*Q* is obtained by using the two dimensions

*FG*

∂ *pi* =1 , (10)

∂ *q i* ), according to the considered func‐

∂ *pi* . (11)

d*F* =1 . (12)

*Q* of a mechanical system, and a unit Poisson bracket be‐

( *f* ). (7)

(*g*). (8)

( *f* , *g*). (9)

$$\frac{d^i p\_i^i}{d\overline{G}} = \frac{\partial \, F}{\partial q^i} \,, \qquad \frac{d^i q^i}{d\overline{G}} = \text{ - } \frac{\partial \, F}{\partial p\_i} \,. \tag{13}$$

Now, *G* is the shift parameter besides of being the conjugate variable to *F*. This also renders the Poisson bracket to the identity

$$\{F,\,\,\,G\} = \frac{d}{dG}\frac{p\_i}{\partial \,p\_i} + \frac{d\,q^i}{dG}\frac{\partial G}{\partial q^i} = \frac{dG}{dG} = \mathbf{1}\,\,. \tag{14}$$

The dynamical systems and vector fields for the motions just defined are

$$\frac{dz}{dG} = X\_F \, \, \, \, \quad \, X\_F = \begin{pmatrix} \frac{\partial F}{\partial p\_i \, \, \, \, \, \, \partial \, \, F} \end{pmatrix} \, \, \, \quad \text{and} \quad \, \frac{dz}{dF} = X\_G \, \, \, \, \, \, \, \, X\_G = \begin{pmatrix} \frac{\partial G}{\partial p\_i \, \, \, \, \, \, \, \, \, \, \partial G} \end{pmatrix} \, \, \, \, \tag{15}$$

Then, the motion along one of the *F* or *G* directions is determined by the corresponding conjugate variable. These vector fields in general are not orthogonal, nor parallel.

If the motion of phase space points is governed by the vector field (15), *F* remains constant because

$$\frac{dF}{d\mathbf{G}} = \frac{\partial \, F}{\partial q^i} \frac{\partial q^i}{\partial \, G} + \frac{\partial \, F}{\partial \, p\_i} \frac{\partial \, p\_i}{\partial \, G} = \frac{\partial \, p\_i}{\partial \, G} \frac{\partial q^i}{\partial \, G} - \frac{\partial q^i}{\partial \, G} \frac{\partial \, p\_i}{\partial \, G} = \mathbf{0} \, \, \tag{16}$$

In contrast, when motion occurs in the *F* direction, by means of Eq. (16), it is the *G* variable the one that remains constant because

$$\frac{dG}{dF} = \frac{\partial G}{\partial q^i}, \frac{\partial q^i}{\partial F} + \frac{\partial G}{\partial p\_i}\frac{\partial p\_i}{\partial F} = \dots \\ \frac{\partial p\_i}{\partial F}\frac{\partial q^i}{\partial F} + \frac{\partial q^i}{\partial F}\frac{\partial p\_i}{\partial F} = 0 \dots \tag{17}$$

Hence, motion originated by the conjugate variables *F* (*z*) and *G*(*z*) occurs on the shells of constant *F* (*z*) or of constant *G*(*z*), respectively.

The divergence of these vector fields is zero,

$$
\nabla \bullet X\_F = -\frac{\partial}{\partial q^i} \frac{\partial F}{\partial p\_i} + \frac{\partial}{\partial p\_i} \frac{\partial F}{\partial q^i} = 0 \ \ \ \ \ \ \nabla \bullet X\_G = \frac{\partial}{\partial q^i} \frac{\partial G}{\partial p\_i} - \frac{\partial}{\partial p\_i} \frac{\partial G}{\partial q^i} = \mathbf{0} \ \ \tag{18}
$$

Thus, the motions associated to each of these conjugate variables preserve the phase space area.

A constant Poisson bracket is related to the constancy of a cross product because

$$\mathbf{X}\_{G} \wedge \mathbf{X}\_{F} = \frac{dz}{dF} \wedge \frac{dz}{dG} = \begin{vmatrix} \stackrel{\scriptstyle \Omega}{q} & \stackrel{\scriptstyle \Omega}{p} & \stackrel{\scriptstyle \Omega}{n} \\ \frac{\partial G}{\partial p} & -\frac{\partial G}{\partial q} & \mathbf{0} \\ -\frac{\partial F}{\partial p} & \frac{\partial F}{\partial q} & \mathbf{0} \end{vmatrix} = \stackrel{\scriptstyle \Omega}{n} \begin{pmatrix} \frac{\partial G}{\partial p} \frac{\partial F}{\partial q} & -\frac{\partial G}{\partial q} \frac{\partial F}{\partial p} \end{pmatrix} = \stackrel{\scriptstyle \Omega}{n} \begin{pmatrix} F \ \ G \end{pmatrix} \tag{19}$$

and the other is

*d df* = *dq df* ∂ ∂ *q* + *dp df* ∂ ∂ *p* + ∂ ∂ *f* = *dz*

*d dg* = *dq dg* ∂ ∂ *q* + *dp dg* ∂ ∂ *p* + ∂ ∂ *g* = *dz*

and

and

L*F* = ∂ *F* ∂ *q* ∂ ∂ *p* - ∂ *F* ∂ *p* ∂

{*F* , *G*}=( ∂ *F*

With these, we have introduced the Liouville type operators

∂

ators and commutators as in Quantum Mechanics.

ieved with the above Liouvillian operators as

∂ *q* ∂ ∂ *p* - ∂ *F* ∂ *p* ∂

∂ *q* = *XF* ∙∇, and L*G* = ∂ *G*

∂ *f* = - L*G*, and ∂

∂

∂

Also, note that for any function *u*(*z*) of a phase space point *z*, we have that

tive of functions vanishes, i.e. the total amount of a function is conserved,

*df* ∙∇ +

*dg* ∙∇ +

L*F* , *u*(*z*) =L*F u*(*z*)= *XF* ∙∇*u*(*z*)= dz

L*G*, *u*(*z*) =L*Gu*(*z*)= *XG* ∙∇*u*(*z*)= *dz*

dependent operators. The formal solutions to these equations are

These are the classical analogues of the quantum evolution equation *d*

∂ *p* ∂ ∂ *q* - ∂ *G* ∂ *q* ∂

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

These are Lie derivatives in the directions of *XF* and *XG*, respectively. These operators gen‐ erate complementary motion of functions in phase space. Note that now, we also have oper‐

Conserved motion of phase space functions moving along the *f* or *g* directions can be ach‐

Indeed, with the help these definitions and of the chain rule, we have that the total deriva‐

∂ ∂ *f* =L*G* +

∂ ∂ *g* =L*F* +

dG ∙∇u(z)= - ∂

*dF* ∙∇*u*(*z*)= - ∂

∂ *f* = *XG* ∙∇ +

∂ *g* = *XF* ∙∇ +

which are the evolution equations for functions along the conjugate directions *f* and *g*.

∂ *q* )*G* = L*F* , *G* =1 . (23)

∂ *g* = - L*F* . (25)

∂ *f* +

∂

∂ *g u*(*z*), (28)

∂ *f u*(*z*) , (29)

*i*ℏ , *H*

^ for time

*dt* = 1

∂ *f* =0 , (26)

∂ *g* =0 . (27)

∂ ∂ *f* = - ∂

∂ ∂ *g* = - ∂ ∂ *g* + ∂

∂ *p* = *XG* ∙∇ . (24)

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9

where *n* ^ is the unit vector normal to the phase space plane. Then, the magnitudes of the vec‐ tor fields and the angle between them changes in such a way that the cross product remains constant when the Poisson bracket is equal to one, i.e. the cross product between conjugate vector fields is a conserved quantity.

The Jacobian for transformations from phase space coordinates to ( *f* , *g*) variables is one for each type of motion:

$$f = \begin{vmatrix} \frac{\partial q}{\partial f} & \frac{\partial p}{\partial f} \\ \frac{\partial q}{\partial g} & \frac{\partial p}{\partial g} \end{vmatrix} = \begin{vmatrix} \frac{\partial G}{\partial p} & -\frac{\partial G}{\partial q} \\ \frac{\partial q}{\partial g} & \frac{\partial p}{\partial g} \end{vmatrix} = \frac{\partial G}{\partial p}\frac{\partial p}{\partial g} + \frac{\partial G}{\partial q}\frac{\partial q}{\partial g} = \frac{dG}{dg} = \mathbf{1} \tag{20}$$

and

$$J = \begin{vmatrix} \frac{\partial q}{\partial f} & \frac{\partial p}{\partial f} \\ \frac{\partial q}{\partial g} & \frac{\partial p}{\partial g} \end{vmatrix} = \begin{vmatrix} \frac{\partial q}{\partial f} & \frac{\partial p}{\partial f} \\ -\frac{\partial F}{\partial p} & \frac{\partial F}{\partial g} \end{vmatrix} = \frac{\partial F}{\partial q} \frac{\partial q}{\partial f} + \frac{\partial F}{\partial p} \frac{\partial p}{\partial f} = \frac{dF}{df} = 1 \,\tag{21}$$

We have seen some properties related to the motion of phase space points caused by conju‐ gate variables.

#### **1.5. Poisson brackets and commutators**

We now consider the use of commutators in the classical realm.

The Poisson bracket can also be written in two ways involving a *commutator.* One form is

$$\{F, G\} = \left(\frac{\partial \, G}{\partial \, p} \frac{\partial}{\partial q} - \frac{\partial \, G}{\partial q} \frac{\partial}{\partial \, p}\right) F = \left[\mathbb{L}\_{G\prime}, F\right] = \mathbf{1}\_{\prime} \tag{22}$$

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables http://dx.doi.org/10.5772/53598 9

and the other is

Thus, the motions associated to each of these conjugate variables preserve the phase space

^ is the unit vector normal to the phase space plane. Then, the magnitudes of the vec‐

tor fields and the angle between them changes in such a way that the cross product remains constant when the Poisson bracket is equal to one, i.e. the cross product between conjugate

The Jacobian for transformations from phase space coordinates to ( *f* , *g*) variables is one for

*dg* =1 , (20)

*df* =1 . (21)

∂ *p* )*F* = L*G*, *F* =1 , (22)

A constant Poisson bracket is related to the constancy of a cross product because

^ *n* ^

∂ *G* ∂ *q* 0

∂ *F* ∂ *q* 0 |=*n* ^( ∂ *G* ∂ *p* ∂ *F* ∂ *q* - ∂ *G* ∂ *q* ∂ *F* ∂ *p* ) =*n*

^ *p*

∂ *G* ∂ *p* -

area.

where *n*

and

gate variables.

each type of motion:

*XG* ∧ *XF* = *dz*

8 Advances in Quantum Mechanics

vector fields is a conserved quantity.

*J* =| ∂ *q* ∂ *f*

*J* =| ∂ *q* ∂ *f*

**1.5. Poisson brackets and commutators**

∂ *q* ∂ *g*

∂ *q* ∂ *g*

∂ *p* ∂ *f*

We now consider the use of commutators in the classical realm.

{*F* , *G*}=( ∂ *G*

∂ *p* ∂ ∂ *q* - ∂ *G* ∂ *q* ∂

∂ *q* ∂ *g* ∂ *p* ∂ *g*

> ∂ *p* ∂ *f*

∂ *F* ∂ *q* |= ∂ *F* ∂ *q* ∂ *q* ∂ *f* + ∂ *F* ∂ *p* ∂ *p* ∂ *f* = *dF*

We have seen some properties related to the motion of phase space points caused by conju‐

The Poisson bracket can also be written in two ways involving a *commutator.* One form is

∂ *p* ∂ *g*

∂ *p* ∂ *f*

∂ *p* ∂ *g*

*dF* ∧ *dz dG* =| *q*

$$\{F\_{\nu}, G\} = \begin{pmatrix} \frac{\partial \cdot F}{\partial q} \frac{\partial}{\partial p} - \frac{\partial \cdot F}{\partial p} \frac{\partial}{\partial q} \end{pmatrix} G = \begin{bmatrix} \mathbb{L}\_{F, \nu} \ G \end{bmatrix} = \mathbf{1} \tag{23}$$

With these, we have introduced the Liouville type operators

$$\mathsf{L}\_{F} = \frac{\partial F}{\partial q} \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } = \mathsf{X}\_{F} \bullet \ \nabla \mathsf{L} \quad \text{ and } \mathsf{L}\_{G} = \frac{\partial G}{\partial p} \frac{\partial}{\partial q} \text{ } \frac{\partial G}{\partial q} \frac{\partial}{\partial p} \text{ } = \mathsf{X}\_{G} \bullet \ \nabla \mathsf{L} \quad \text{and } \nabla \cdot \mathsf{L} = \mathsf{L}\_{F} \bullet \ \nabla \mathsf{L} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial$$

These are Lie derivatives in the directions of *XF* and *XG*, respectively. These operators gen‐ erate complementary motion of functions in phase space. Note that now, we also have oper‐ ators and commutators as in Quantum Mechanics.

Conserved motion of phase space functions moving along the *f* or *g* directions can be ach‐ ieved with the above Liouvillian operators as

$$\frac{\partial}{\partial f} = -\mathsf{L}\_{G\prime} \quad \text{and} \qquad \frac{\partial}{\partial g} = -\mathsf{L}\_{F} \quad . \tag{25}$$

Indeed, with the help these definitions and of the chain rule, we have that the total deriva‐ tive of functions vanishes, i.e. the total amount of a function is conserved,

$$\frac{d}{d\boldsymbol{f}} = \frac{dq}{d\boldsymbol{f}} \frac{\partial}{\partial q} + \frac{dp}{d\boldsymbol{f}} \frac{\partial}{\partial p} + \frac{\partial}{\partial f} = \frac{dz}{d\boldsymbol{f}} \bullet \nabla + \frac{\partial}{\partial f} = \boldsymbol{X}\_G \bullet \nabla + \frac{\partial}{\partial f} = \boldsymbol{\L}\_G + \frac{\partial}{\partial f} = \boldsymbol{-\frac{\partial}{\partial f}} + \frac{\partial}{\partial f} = \boldsymbol{0} \,,\tag{26}$$

and

$$\frac{d}{d\mathbf{g}} = \frac{d\mathbf{q}}{d\mathbf{g}} \frac{\partial}{\partial q} + \frac{d\mathbf{p}}{d\mathbf{g}} \frac{\partial}{\partial p} + \frac{\partial}{\partial g} = \frac{dz}{d\mathbf{g}} \bullet \nabla + \frac{\partial}{\partial g} = X\_F \bullet \nabla + \frac{\partial}{\partial g} = \mathsf{L}\_F + \frac{\partial}{\partial g} = \mathsf{L}\_{\overline{\partial g}} + \frac{\partial}{\partial g} = \mathsf{0} \ . \tag{27}$$

Also, note that for any function *u*(*z*) of a phase space point *z*, we have that

$$\mathbb{L}\_{F}\left[\mathbb{L}\_{F}\left.\mu(z)\right]\right] = \mathbb{L}\_{F}\mu(z) = X\_{F}\bullet\nabla\,\mu(z) = \frac{dx}{dG}\bullet\nabla\,\mathbf{u}(z) = -\frac{\partial}{\partial g}\mu(z),\tag{28}$$

and

$$\mathbb{L}\_{G\prime}\ u(z)\mathbb{I} = \mathbb{L}\_{G}u(z) = X\_{G} \bullet \nabla\ u(z) = \frac{dz}{d\mathbb{F}} \bullet \nabla\ u(z) = -\frac{\partial}{\partial f}u(z) \,,\tag{29}$$

which are the evolution equations for functions along the conjugate directions *f* and *g*. These are the classical analogues of the quantum evolution equation *d dt* = 1 *i*ℏ , *H* ^ for time dependent operators. The formal solutions to these equations are

$$u(z; \mathcal{g}) = e^{\cdot \mathcal{g} \mathcal{L}\_F} u(z) \quad \quad \text{and} \quad u(z; f) = e^{-f^{\perp\_{\mathcal{L}\_G}}} u(z) \,. \tag{30}$$

This is a relationship that indicates how to translate the function *F* (*z*) as an operator. When this equality is acting on the number one, we arrive at the translation property for *F* as a function

i.e., up to an additive constant, *f* is the value of *F* (*z*) itself, one can be replaced by the other and actually they are the same object, with *f* the classical analogue of the spectrum of a

Continuing in a similar way, we can obtain the relationships shown in the following diagram

*d*

This implies that

quantum operator.

**Diagram 1.**

*F* (*z*; *f* )=(*e f* L*G F* (*z*)) = *F* (*z*) *e f* L*G* 1+ *f e f* L*G*1= *F*(*z*) + *f* . (37)

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

*df F*(*z*; *f* )=1 , (38)

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11

With these equations, we can now move a function *u*(*z*) on *T \* Q* in such a way that the points of their support move according to the dynamical systems Eqs. (15) and the total amount of *u* is conserved.

#### **1.6. The commutator as a derivation and its consequences**

As in quantum theory, we have found commutators and there are many properties based on them, taking advantage of the fact that a commutator is a derivation.

Since the commutator is a derivation, for conjugate variables *F* (*z*) and *G*(*z*) we have that, for integer *n*,

$$\left[\mathsf{L}\_{G,}^{\tiny\tiny\raisebox{0.75pt}{ $\mathsf{L}\_{G}$ }}\,\middle|\,\mathrm{F}\right] \equiv \mathsf{n}\,\mathsf{L}\_{G}^{\tiny\raisebox{0.75pt}{ $\mathsf{L}\_{G}$ }}\,\mathsf{A}\qquad\left[\mathsf{L}\_{G,}\,\mathsf{F}\,\middle|\,\mathrm{F}\right] \equiv \mathsf{n}\,\mathsf{F}^{\mathop\rule{0.75pt}{ $\mathsf{L}\_{F}$ }}\,\mathsf{G}\,\middle|\,\mathrm{G}\right] \equiv \mathsf{n}\,\mathsf{L}\_{F}^{\mathop\rule{0.75pt}{ $\mathsf{L}\_{F}$ }}\,\mathsf{A}\qquad\left[\mathsf{L}\_{F},\,\mathsf{G}\,\middle|\,\mathrm{F}\right] \equiv \mathsf{n}\,\mathsf{G}^{\mathop\rule{0.75pt}{ $\mathsf{L}\_{F}$ }}\,\mathsf{A}\tag{31}$$

Based on the above equalities, we can get translation relationships for functions on *T \* Q*. We first note that, for a holomorphic function *u*(*x*)= ∑ *n*=0 *∞ unx n*,

$$\left[\mu\left(\mathsf{L}\_{\mathsf{G}}\right),\,\,F\right] = \left[\sum\_{n=0}^{\mathsf{m}}\mu\_{n}\mathsf{L}\_{\mathsf{G},\prime}^{\mathsf{m}}\,\,\_{\mathsf{F}}\right] = \sum\_{n=0}^{\mathsf{m}}\mu\mu\_{n}\mathsf{L}\_{\mathsf{G}}^{\mathsf{m}-1} = \mu^{\prime}\left(\mathsf{L}\_{\mathsf{G}}\right)\,\,. \tag{32}$$

In particular, we have that

$$\begin{bmatrix} \int e^{f\,\mathsf{L}\_{G}} \, . \, F \end{bmatrix} = f \, e^{f\,\mathsf{L}\_{G}} \, . \tag{33}$$

Then, *e f* L*G* is the eigenfunction of the commutator ∙, *F* with eigenvalue *f* .

From Eq. (32), we find that

$$
u(\mathsf{L}\_G) \mathcal{F} \text{ -- } \mathsf{F} \mathcal{u}(\mathsf{L}\_G) \coloneqq \mathsf{u}^\top(\mathsf{L}\_G).\tag{34}$$

But, if we multiply by *u* -1(L*G*) from the right, we arrive to

$$
\mu\left(\mathsf{L}\_{G}\right)\mathsf{F}\,\mu^{-1}\left(\mathsf{L}\_{G}\right) = \mathsf{F} + \mu^{\top}\left(\mathsf{L}\_{G}\right)\mu^{-1}\left(\mathsf{L}\_{G}\right).\tag{35}
$$

This is a generalized version of a shift of *F* , and the classical analogue of a generalization of the quantum Weyl relationship. A simple form of the above equality, a familiar form, is ob‐ tained with the exponential function, i.e.

$$\text{If } e^{f\perp\_{\mathcal{C}}} \mathcal{F} e^{-f\perp\_{\mathcal{C}}} = \mathcal{F} + f \text{ .} \tag{36}$$

This is a relationship that indicates how to translate the function *F* (*z*) as an operator. When this equality is acting on the number one, we arrive at the translation property for *F* as a function

$$F\left(z;f\right) = \left(e^{f\left\lfloor\operatorname{L}\_{\mathbb{C}}\right\rfloor}F\left(z\right)\right) = F\left(z\right)\left(e^{f\left\lfloor\operatorname{L}\_{\mathbb{C}}\right\rfloor}1 + f\right)e^{f\left\lfloor\operatorname{L}\_{\mathbb{C}}\right\rfloor}1 = F\left(z\right) + f\left.\right.\tag{37}$$

This implies that

*u*(*z*; *g*)=*e*-*g*L*F u*(*z*) , and *u*(*z*; *f* )=*e*- *f* L*Gu*(*z*). (30)

points of their support move according to the dynamical systems Eqs. (15) and the total

As in quantum theory, we have found commutators and there are many properties based on

Since the commutator is a derivation, for conjugate variables *F* (*z*) and *G*(*z*) we have that, for

*n* , *G* =*n* L*F*

*n*=0 *∞ unx n*,

, L*F*

Based on the above equalities, we can get translation relationships for functions on *T \**

*n* , *F* = ∑ *n*=0 ∞ *nun*L*G n*-1 =*u* ' *Q* in such a way that the

*n*-1 , L*F* , *G n* =*n G n*-1 . (31)

(L*G*) . (32)

(L*G*). (34)

(L*G*)*u* -1(L*G*). (35)

*e f* L*G F e*- *f* L*G* = *F* + *f* . (36)

*e f* L*G*, *F* = *f e f* L*G* . (33)

*Q*. We

With these equations, we can now move a function *u*(*z*) on *T \**

them, taking advantage of the fact that a commutator is a derivation.

**1.6. The commutator as a derivation and its consequences**

*n*-1 , L*G*, *F n* =*n F n*-1

*u*(L*G*), *F* = ∑

But, if we multiply by *u* -1(L*G*) from the right, we arrive to

tained with the exponential function, i.e.

*n*=0 ∞ *un*L*G*

Then, *e f* L*G* is the eigenfunction of the commutator ∙, *F* with eigenvalue *f* .

*u*(L*G*)*F* - *Fu*(L*G*) =*u* '

This is a generalized version of a shift of *F* , and the classical analogue of a generalization of the quantum Weyl relationship. A simple form of the above equality, a familiar form, is ob‐

*u*(L*G*)*F u* -1(L*G*) = *F* + *u* '

first note that, for a holomorphic function *u*(*x*)= ∑

amount of *u* is conserved.

10 Advances in Quantum Mechanics

integer *n*,

L*G*

*n* , *F* =*n* L*G*

In particular, we have that

From Eq. (32), we find that

$$\mathbf{1} \frac{d}{d\mathbf{f}} \mathbf{F}\left(z; f\right) = \mathbf{1} \,, \tag{38}$$

i.e., up to an additive constant, *f* is the value of *F* (*z*) itself, one can be replaced by the other and actually they are the same object, with *f* the classical analogue of the spectrum of a quantum operator.

Continuing in a similar way, we can obtain the relationships shown in the following diagram

**Diagram 1.**

where the constant *s* has units of action, length times momentum, the same units as the quantum constant ℏ.

The eigenvectors of the position, momentum and energy operators have been used to pro‐ vide a representation of wave functions and of operators. So, in general, the eigenvectors

^ provide with a set of vectors for a represen‐

. (40)

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13

^ ) , (41)

^ . (42)

^ ). (43)

^ ). (44)

. The usual shift

^ + *f* . (45)

^ *n*-1

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

*unz n* we have that

^ and *G*

With the help of the properties of commutators between operators, we can see that

, *F* ^ , *G* ^ *n* =*i*ℏ*G*

*n*=0 *∞*

^) , *F*

^ , 1

A set of equalities is obtained from Eq. (43) by first writing them in expanded form as

^), and *F*

^), and *u*

^ + *g*, and *F*

These are a set of generalized shift relationships for the operators *G*

relationships are obtained when *u*(*x*) is the exponential function, i.e.

^ , *u* ^(*G* ^ ) =*i*ℏ*u* ^' (*G*

i.e., the commutators behave as derivations with respect to operators. In an abuse of nota‐

*i*ℏ *F* ^

> ^ *u* ^(*G* ^ ) - *u* ^(*G* ^ )*F* ^ =*i*ℏ*u* ^' (*G*

^-1(*G* ^ )*F* ^ *u* ^(*G* ^ )= *F*

^( *f* ): =*eif G*

^ /ℏ*F* ^ *e*-*if G* ^ /ℏ = *F*

^ + *i*ℏ*u*

^-1(*G* ^ )*u* ^' (*G*

^ and *F* ^

Next, we multiply these equalities by the inverse operator to the right or to the left in order

We can take advantage of this fact and derive the quantum versions of the equalities found

, ∙ = *d* ∙ *dG*

tation of dynamical quantities like the wave functions *f* | *ψ* and *g* | *ψ* .

^ =*i*ℏ*F* ^ *n*-1

^ =*i*ℏ*u* ^' (*F*

> ^ = *d* ∙ *d F*

Hence, for a holomorphic function *u*(*z*)= ∑

*u* ^(*F* ^), *G*

tion, we have that

in the classical realm.

to obtain

*u* ^(*F* ^)*G* ^ *u* ^-1(*F* ^)=*G*

*G*

^ (*g*): =*e*-*ig F*

^ /ℏ*G* ^ *eig F* ^ /ℏ =*G*

*u* ^(*F* ^)*G* ^ - *G* ^ *u* ^(*F* ^)=*i*ℏ*u* ^' (*F*

*F* ^ *n*, *G*

1 *i*ℏ ∙, *G*

^ + *i*ℏ*u* ^' (*F* ^)*u* ^-1(*F*

Some of the things to note are:

The operator *e g*L*F* is the eigenoperator of the commutator ∙, *G* and can be used to gener‐ ate translations of *G*(*z*) as an operator or as a function. This operator is also the propagator for the evolution of functions along the *g* direction. The variable *g* is more than just a shift parameter; it actually labels the values that *G*(*z*) takes, the classical analogue of the spec‐ trum of a quantum operator.

The operators L*F* and *G*(*z*) are also a pair of conjugate operators, as well as the pair L*G* and *F* (*z*).

But L*F* commutes with *F* (*z*) and then it cannot be used to translate functions of *F* (*z*), *F* (*z*) is a conserved quantity when motion occurs along the *G*(*z*) direction.

The eigenfunction of L*F* , ∙ and of *s*L*F* is *e fG*(*z*)/*s* and this function can be used to shift L*F* as an operator or as a function.

The variable *f* is more than just a parameter in the shift of *s*L*F* , it actually is the value that *s*L*F* can take, the classical analogue of the spectra of a quantum operator.

The steady state of L*F* is a function of *F* (*z*), but *e gF* (*z*)/*s* is an eigenfunction of L*G* and of L*G*, ∙ and it can be used to translate L*G*.

These comments involve the left hand side of the above diagram. There are similar conclu‐ sions that can be drawn by considering the right hand side of the diagram.

Remember that the above are results valid for classical systems. Below we derive the corre‐ sponding results for quantum systems.

### **2. Quantum systems**

We now derive the quantum analogues of the relationships found in previous section. We start with a Hilbert space H of wave functions and two conjugate operators *F* ^ and *G* ^ acting on vectors in H, and with a constant commutator between them

$$\begin{aligned} \text{[}\stackrel{\wedge}{F}, \stackrel{\wedge}{G}] &= i\hbar \, , \end{aligned} \tag{39}$$

together with the domain D=D(*F* ^ *G* ^ )∩D(*G* ^ *F* ^) in which the commutator holds. Examples of these operators are coordinate *Q* ^ and momentum *P* ^ operators, energy *H* ^ and time *T* ^ opera‐ tors, creation *a* ^† and annihilation *a* ^ operators.

The eigenvectors of the position, momentum and energy operators have been used to pro‐ vide a representation of wave functions and of operators. So, in general, the eigenvectors | *f* and | *g* of the conjugate operators *F* ^ and *G* ^ provide with a set of vectors for a represen‐ tation of dynamical quantities like the wave functions *f* | *ψ* and *g* | *ψ* .

With the help of the properties of commutators between operators, we can see that

$$[\stackrel{\frown}{F}^n, \stackrel{\frown}{G}] = i\hbar \hat{F}^{n-1}, \qquad [\stackrel{\frown}{F}\_{\prime}, \stackrel{\frown}{G}^n] = i\hbar \hat{G}^{n-1}.\tag{40}$$

Hence, for a holomorphic function *u*(*z*)= ∑ *n*=0 *∞ unz n* we have that

where the constant *s* has units of action, length times momentum, the same units as the

The operator *e g*L*F* is the eigenoperator of the commutator ∙, *G* and can be used to gener‐ ate translations of *G*(*z*) as an operator or as a function. This operator is also the propagator for the evolution of functions along the *g* direction. The variable *g* is more than just a shift parameter; it actually labels the values that *G*(*z*) takes, the classical analogue of the spec‐

The operators L*F* and *G*(*z*) are also a pair of conjugate operators, as well as the pair L*G*

But L*F* commutes with *F* (*z*) and then it cannot be used to translate functions of *F* (*z*), *F* (*z*) is

The eigenfunction of L*F* , ∙ and of *s*L*F* is *e fG*(*z*)/*s* and this function can be used to shift L*F* as

The variable *f* is more than just a parameter in the shift of *s*L*F* , it actually is the value that

These comments involve the left hand side of the above diagram. There are similar conclu‐

Remember that the above are results valid for classical systems. Below we derive the corre‐

We now derive the quantum analogues of the relationships found in previous section. We

^

start with a Hilbert space H of wave functions and two conjugate operators *F*

*F* ^ , *G*

^ operators.

^ and momentum *P*

on vectors in H, and with a constant commutator between them

^ *G* ^ )∩D(*G* ^ *F* is an eigenfunction of L*G* and of

^ and *G*

^ and time *T*

^ =*i*ℏ , (39)

^) in which the commutator holds. Examples of

operators, energy *H*

^ acting

^ opera‐

a conserved quantity when motion occurs along the *G*(*z*) direction.

*s*L*F* can take, the classical analogue of the spectra of a quantum operator.

sions that can be drawn by considering the right hand side of the diagram.

The steady state of L*F* is a function of *F* (*z*), but *e gF* (*z*)/*s*

L*G*, ∙ and it can be used to translate L*G*.

sponding results for quantum systems.

**2. Quantum systems**

together with the domain D=D(*F*

and annihilation *a*

these operators are coordinate *Q*

^†

tors, creation *a*

quantum constant ℏ.

12 Advances in Quantum Mechanics

Some of the things to note are:

trum of a quantum operator.

an operator or as a function.

and *F* (*z*).

$$\begin{array}{ll} \tiny \begin{array}{l} \widehat{\ulcorner} \widehat{\urcorner} \end{array} \widehat{\ulcorner} \end{array} \begin{array}{l} \widehat{\ulcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \end{array} \tag{41}$$

i.e., the commutators behave as derivations with respect to operators. In an abuse of nota‐ tion, we have that

$$\begin{array}{ll}\frac{1}{\sqrt{\hbar}}\mathsf{f}\bullet\ \bigwedge^{\wedge}\mathsf{G}\begin{array}{c}\mathsf{I}\end{array}\mathsf{J}=\frac{d\,\mathsf{A}}{d\,\mathsf{F}}\end{array},\qquad\qquad\qquad\qquad\qquad\qquad\frac{1}{\sqrt{\hbar}}\mathsf{f}\begin{array}{c}\bigwedge^{\wedge}\mathsf{F}\ \rightarrow\mathsf{J}=\frac{d\,\mathsf{A}}{d\,\mathsf{G}}\end{array},\tag{42}$$

We can take advantage of this fact and derive the quantum versions of the equalities found in the classical realm.

A set of equalities is obtained from Eq. (43) by first writing them in expanded form as

$$
\hat{\mathfrak{u}}(\stackrel{\wedge}{F})\hat{\mathbf{G}} \cdot \stackrel{\wedge}{G}\hat{\mathfrak{u}}(\stackrel{\wedge}{F}) = i\hbar \hat{\mathfrak{u}}^{\wedge}(\stackrel{\wedge}{F}), \qquad \text{and} \qquad \hat{\mathcal{F}}\hat{\mathfrak{u}}(\stackrel{\wedge}{G}) \cdot \stackrel{\wedge}{\hat{\mathfrak{u}}}(\stackrel{\wedge}{G})\hat{\mathbf{F}} = i\hbar \hat{\mathfrak{u}}^{\wedge}(\stackrel{\wedge}{G}).\tag{43}
$$

Next, we multiply these equalities by the inverse operator to the right or to the left in order to obtain

$$
\stackrel{\scriptstyle\wedge}{u}(\stackrel{\scriptstyle\wedge}{F})\stackrel{\scriptstyle\wedge}{G}\stackrel{\scriptstyle\wedge}{u}^{-1}(\stackrel{\scriptstyle\wedge}{F}) = \stackrel{\scriptstyle\wedge}{G} + i\hbar\stackrel{\scriptstyle\wedge}{u}(\stackrel{\scriptstyle\wedge}{F})\stackrel{\scriptstyle\wedge}{u}^{-1}(\stackrel{\scriptstyle\wedge}{F}), \quad \text{and} \quad \stackrel{\scriptstyle\wedge}{u}^{-1}(\stackrel{\scriptstyle\wedge}{G})\stackrel{\scriptstyle\wedge}{F}\stackrel{\scriptstyle\wedge}{u}(\stackrel{\scriptstyle\wedge}{G}) = \stackrel{\scriptstyle\wedge}{F} + i\hbar\stackrel{\scriptstyle\wedge}{u}^{-1}(\stackrel{\scriptstyle\wedge}{G})\stackrel{\scriptstyle\wedge}{u}^{\wedge}(\stackrel{\scriptstyle\wedge}{G}).\tag{44}
$$

These are a set of generalized shift relationships for the operators *G* ^ and *F* ^ . The usual shift relationships are obtained when *u*(*x*) is the exponential function, i.e.

$$\stackrel{\wedge}{G}(\mathcal{G}) \colon= e^{-i\boldsymbol{\varrho}^{\hat{\sf F}}/\hbar} \stackrel{\wedge}{G} e^{i\boldsymbol{\varrho}^{\hat{\sf F}}/\hbar} = \stackrel{\wedge}{G} + \stackrel{\wedge}{g}, \qquad \text{and} \quad \stackrel{\wedge}{F}(f) \colon= e^{i\boldsymbol{f}^{\hat{\sf G}}/\hbar} \stackrel{\wedge}{F} e^{-i\boldsymbol{f}^{\hat{\sf G}}/\hbar} = \stackrel{\wedge}{F} + f \,. \tag{45}$$

Now, as in Classical Mechanics, the commutator between two operators can be seen as two different derivatives introducing quantum dynamical system as

$$\frac{\stackrel{\bigwedge}{d\mathbb{P}}(f)}{\stackrel{\bigwedge}{d\mathbb{P}}} = \text{ } \frac{\stackrel{\bigwedge}{\otimes}\stackrel{\bigwedge}{\mathbb{Q}}\stackrel{\bigwedge}{\mathbb{P}}}{\stackrel{\bigwedge}{\mathbb{Q}}\!\!} = \frac{1}{\stackrel{\bigwedge}{\mathbb{H}}} \stackrel{\bigwedge}{\mathbb{P}} \begin{pmatrix} f \\ f \end{pmatrix} \stackrel{\bigwedge}{\mathbb{G}} \begin{pmatrix} \stackrel{\bigwedge}{\mathbb{Q}}\stackrel{\bigwedge}{\mathbb{P}}\!\!}{\stackrel{\bigwedge}{\mathbb{Q}}\!\!} \end{pmatrix} \qquad \frac{\stackrel{\bigwedge}{d\mathbb{Q}}(f)}{\stackrel{\bigwedge}{\mathbb{P}}} = \frac{\stackrel{\bigwedge}{\mathbb{G}}\stackrel{\bigwedge}{\mathbb{Q}}\stackrel{\bigwedge}{\mathbb{P}}\stackrel{\bigwedge}{\mathbb{P}}}{\stackrel{\bigwedge}{\mathbb{P}}\mathbb{P}} = \frac{1}{\stackrel{\bigwedge}{\mathbb{R}}} \!\!} \stackrel{\bigwedge}{\mathbb{Q}}(f) \stackrel{\bigwedge}{\mathbb{Q}} \begin{pmatrix} \stackrel{\bigwedge}{\mathbb{Q}}\stackrel{\bigwedge}{\mathbb{P}}\stackrel{\bigwedge}{\mathbb{P}}\mathbb{P}}{\stackrel{\bigwedge}{\mathbb{Q}}\!} \end{pmatrix} \tag{46}$$

$$\frac{\stackrel{\frown}{AD}(\mathcal{G})}{\stackrel{\frown}{dg}} = \frac{\stackrel{\frown}{\partial}\stackrel{\frown}{F}(\stackrel{\frown}{Q},\stackrel{\frown}{P})}{\stackrel{\frown}{\partial}Q} = \frac{1}{\stackrel{\frown}{\hbar}\!\!\!} \stackrel{\frown}{F}(\stackrel{\frown}{Q},\stackrel{\frown}{P}),\ \stackrel{\frown}{P}(\mathcal{g})\mathbf{I},\qquad\text{and}\qquad\frac{\stackrel{\frown}{dQ}(\mathcal{g})}{\stackrel{\frown}{dg}} = -\frac{\stackrel{\frown}{\partial}\stackrel{\frown}{F}(\stackrel{\frown}{Q},\stackrel{\frown}{P})}{\stackrel{\frown}{\partial}P} = \frac{1}{\stackrel{\frown}{\hbar}\!\!} \!\!\!\!\!\!\!\!\!\!\!\!\!\begin{B}\stackrel{\frown}{Q}(\stackrel{\frown}{Q},\stackrel{\frown}{Q}),\ \stackrel{\frown}{Q}(\mathcal{g})\mathbf{I},\tag{47}$$

where

$$
\stackrel{\scriptstyle \hat{\Lambda}}{P}(f) = e^{\circ \stackrel{\scriptstyle \hat{G}}{} \stackrel{\scriptstyle \hat{G}}{/\hbar} \stackrel{\scriptstyle \hat{\Lambda}}{P} e^{\circ \stackrel{\scriptstyle \hat{G}}{} \stackrel{\scriptstyle \hat{G}}{/\hbar} \stackrel{\scriptstyle \hat{\Lambda}}{}{/\hbar}}, \qquad \stackrel{\scriptstyle \hat{\Lambda}}{Q}(f) = e^{\circ \stackrel{\scriptstyle \hat{G}}{} \stackrel{\scriptstyle \hat{G}}{} / \hbar} \stackrel{\scriptstyle \hat{\Lambda}}{Q} e^{\circ \stackrel{\scriptstyle \hat{G}}{} \stackrel{\scriptstyle \hat{G}}{} / \hbar}, \tag{48}
$$

There are many equalities that can be obtained as in the classical case. The following dia‐

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

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15

gram shows some of them:

**Diagram 2.**

$$
\stackrel{\frown}{P}(\mathbf{g}) = e^{-i\mathbf{g}\cdot\hat{\mathbf{F}}/\hbar} \stackrel{\frown}{P} e^{i\mathbf{f}\cdot\hat{\mathbf{F}}/\hbar}, \qquad \text{and} \qquad \stackrel{\frown}{Q}(\mathbf{g}) = e^{-i\mathbf{g}\cdot\hat{\mathbf{F}}/\hbar} \stackrel{\frown}{Q} e^{i\mathbf{g}\cdot\hat{\mathbf{F}}/\hbar}. \tag{49}
$$

These equations can be written in the form of a set of quantum dynamical systems

$$\stackrel{\hat{d}\cdot\hat{z}}{\frac{d\hat{\mathcal{L}}}{d\hat{f}}} = \stackrel{\bigwedge}{\mathcal{X}}\_{G\text{-}f}, \qquad \stackrel{\bigwedge}{\mathcal{X}}\_{G} = \begin{pmatrix} \stackrel{\wedge}{\partial\!\hat{Q}}\_{\mathcal{P}} & \stackrel{\wedge}{\partial\!\hat{Q}}\_{\mathcal{P}} \end{pmatrix}, \qquad \qquad \stackrel{\hat{d}\cdot\hat{z}}{\frac{d\hat{z}}{d\mathbf{g}}} = \stackrel{\wedge}{\hat{\mathcal{X}}}\_{F\text{-}f}, \qquad \qquad \stackrel{\bigwedge}{\hat{\mathcal{X}}}\_{F} = \begin{pmatrix} \stackrel{\wedge}{\partial\!\hat{F}}\_{\overline{\mathcal{Q}}} & \stackrel{\wedge}{\partial\!\hat{Q}}\_{\overline{\mathcal{Q}}} \end{pmatrix},\tag{50}$$

where *z* ^ =(*Q* ^ , *P* ^).

The inner product between the operator vector fields is

$$\stackrel{\frown}{X}\stackrel{+}{F}\_{F}\bullet\stackrel{\frown}{X}\_{F} = \begin{pmatrix} \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} & \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} \\ \stackrel{\frown}{\partial P}\prime & \stackrel{\frown}{\partial Q} \end{pmatrix} \stackrel{\sharp}{\bullet} \bullet \begin{pmatrix} \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} & \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} \\ \stackrel{\frown}{\partial P}\prime & \stackrel{\frown}{\partial Q} \end{pmatrix} \equiv \begin{pmatrix} \stackrel{\frown}{d\,\tilde{Q}} \\ \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} \end{pmatrix} + \begin{pmatrix} \stackrel{\frown}{d\,\tilde{P}} \\ \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} \end{pmatrix} \tag{51}$$

where (*d l* ^ *F* )2≔(*dQ* ^ )2 + (*dP* ^)2 , evaluated along the *g* direction, is the quantum analogue of the square of the line element (*dlF* )2 =(*dq*)2 + (*dp*)2 .

We can define many of the classical quantities but now in the quantum realm. Liouville type operators are

$$\stackrel{\scriptstyle \mathsf{A}}{\mathsf{L}}\_{F} \coloneqq \frac{1}{i\hbar} \stackrel{\scriptstyle \mathsf{f}}{\mathsf{f}}\_{F} \bullet \mathsf{J}\_{\prime} \qquad \text{and} \quad \stackrel{\scriptstyle \mathsf{A}}{\mathsf{L}}\_{G} \coloneqq \frac{1}{i\hbar} \mathsf{I} \bullet \stackrel{\scriptstyle \mathsf{A}}{\mathsf{G}} \mathsf{J} \,. \tag{52}$$

These operators will move functions of operators along the conjugate directions *G* ^ or *F* ^ , re‐ spectively. This is the case when *G* ^ is the Hamiltonian *H* ^ of a physical system, a case in which we get the usual time evolution of operator.

There are many equalities that can be obtained as in the classical case. The following dia‐ gram shows some of them:

**Diagram 2.**

 (49)

^ ) , (50)

, (51)

^ . (52)

^ of a physical system, a case in

^ or *F* ^ , re‐

^) , *dQ*

^(*g*) , and *dQ*

/ℏ , *Q*

/ℏ , and *Q*

These equations can be written in the form of a set of quantum dynamical systems

^ ) , *d z*

^ *dg* = *X* ^

.

We can define many of the classical quantities but now in the quantum realm.  This is the case when *G*

Note that the conclusions mentioned at the end of the previous section for classical systems also hold in the quantum realm.

Next, we illustrate the use of these ideas with a simple system.

#### **3. Time evolution using energy and time eigenstates**

As a brief application of the abovee ideas, we show how to use the energy-time coordinates and eigenfunctions in the reversible evolution of probability densities.

Earlier, there was an interest on the classical and semi classical analysis of energy transfer in molecules. Those studies were based on the quantum procedure of expanding wave func‐ tions in terms of energy eigenstates, after the fact that the evolution of energy eigenstates is quite simple in Quantum Mechanics because the evolution equation for a wave function *i*ℏ ∂ ∂ *t* |*ψ*> =*H* ^ |*ψ*> is linear and contains the Hamiltonian operator. In those earlier calcula‐ tions, an attempt to use the eigenfunctions of a complex classical Liouville operator was made [5-8]. The results in this chapter show that the eigenfunction of the Liouville operator L*H* is *e gT* (*z*) and that it do not seems to be a good set of functions in terms of which any other function can be written, as is the case for the eigenfunctions of the Hamiltonian operator in Quantum Mechanics. In this section, we use the time eigenstates instead.

With energy-time eigenstates the propagation of classical densities is quite simple. In order to illustrate our procedure, we will apply it to the harmonic oscillator with Hamiltonian giv‐ en by (we will use dimensionless units)

$$H(z) = \frac{p^2}{2} + \frac{q^2}{2} \,. \tag{53}$$

*p* =*q* tan (*t* + *π*

is a periodic system, so we will only consider one period in time.

2 ), or *q* = *p* cot (*t* + *π*

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

These are just straight lines passing through the origin, equivalent to the polar coordinates. The value of time on these points is *t*, precisely. In Fig. 1, we show both coordinate systems, the phase space coordinates (*q*, *p*), and the energy time coordinates (*E*, *t*) on the plane. This

**Figure 1.** Two conjugate coordinate systems for the classical harmonic oscillator in dimensionless units. Blue and black

At this point, there are two options for time curves. Both options will cover the plane and we can distinguish between the regions of phase space with negative or positive momen‐ tum. One is to use half lines and *t* in the range from -*π* to *π*, with the curve *t* =0 coinciding with the positive *p* axes. The other option is to use the complete curve including positive

lines correspond to the (*q*, *p*) coordinates and the red and green curves to the (*E*, *t*) coordinates.

2 ) . (56)

http://dx.doi.org/10.5772/53598

17

Given and energy scaling parameter *Es* and the frequency *ω* of the harmonic oscillator, the remaining scaling parameters are

$$p\_s = \sqrt{mE\_s} \quad , \qquad q\_s = \sqrt{\frac{E\_s}{m\omega^2}} \quad , \qquad t\_s = \frac{1}{\omega} \quad . \tag{54}$$

We need to define time eigensurfaces for our calculations. The procedure to obtain them is to take the curve *q* =0 as the zero time curve. The forward and backward propagation of the zero time curve generates the time coordinate system in phase space. The trajectory generat‐ ed with the harmonic oscillator Hamiltonian is

$$q(t) = \sqrt{2E} \cos\left(t + \frac{\pi}{2}\right), \qquad p(t) = \sqrt{2E} \sin\left(t + \frac{\pi}{2}\right). \tag{55}$$

With the choice of phase we have made, *q* =0 when *t* =0, which is the requirement for an ini‐ tial time curve. Then, the equation for the time curve is

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables http://dx.doi.org/10.5772/53598 17

$$p = q \tan\left(t + \frac{\pi}{2}\right), \quad \text{or} \quad q = p \cot\left(t + \frac{\pi}{2}\right). \tag{56}$$

These are just straight lines passing through the origin, equivalent to the polar coordinates. The value of time on these points is *t*, precisely. In Fig. 1, we show both coordinate systems, the phase space coordinates (*q*, *p*), and the energy time coordinates (*E*, *t*) on the plane. This is a periodic system, so we will only consider one period in time.

Note that the conclusions mentioned at the end of the previous section for classical systems

As a brief application of the abovee ideas, we show how to use the energy-time coordinates

Earlier, there was an interest on the classical and semi classical analysis of energy transfer in molecules. Those studies were based on the quantum procedure of expanding wave func‐ tions in terms of energy eigenstates, after the fact that the evolution of energy eigenstates is quite simple in Quantum Mechanics because the evolution equation for a wave function

tions, an attempt to use the eigenfunctions of a complex classical Liouville operator was made [5-8]. The results in this chapter show that the eigenfunction of the Liouville operator

function can be written, as is the case for the eigenfunctions of the Hamiltonian operator in

With energy-time eigenstates the propagation of classical densities is quite simple. In order to illustrate our procedure, we will apply it to the harmonic oscillator with Hamiltonian giv‐

2 + *q* 2

Given and energy scaling parameter *Es* and the frequency *ω* of the harmonic oscillator, the

We need to define time eigensurfaces for our calculations. The procedure to obtain them is to take the curve *q* =0 as the zero time curve. The forward and backward propagation of the zero time curve generates the time coordinate system in phase space. The trajectory generat‐

With the choice of phase we have made, *q* =0 when *t* =0, which is the requirement for an ini‐

*mω* 2 , *ts* = 1

2 ) , *p*(*t*)= 2*E*sin (*t* + *π*

^ |*ψ*> is linear and contains the Hamiltonian operator. In those earlier calcula‐

and that it do not seems to be a good set of functions in terms of which any other

2 . (53)

*ω* . (54)

2 ) . (55)

also hold in the quantum realm.

16 Advances in Quantum Mechanics

*i*ℏ ∂ ∂ *t*

L*H* is *e gT* (*z*)

en by (we will use dimensionless units)

remaining scaling parameters are

ed with the harmonic oscillator Hamiltonian is

*q*(*t*)= 2*E*cos (*t* + *π*

tial time curve. Then, the equation for the time curve is

Next, we illustrate the use of these ideas with a simple system.

**3. Time evolution using energy and time eigenstates**

and eigenfunctions in the reversible evolution of probability densities.

Quantum Mechanics. In this section, we use the time eigenstates instead.

*H* (*z*)= *p* 2

*ps* = *mEs* , *qs* = *Es*

**Figure 1.** Two conjugate coordinate systems for the classical harmonic oscillator in dimensionless units. Blue and black lines correspond to the (*q*, *p*) coordinates and the red and green curves to the (*E*, *t*) coordinates.

At this point, there are two options for time curves. Both options will cover the plane and we can distinguish between the regions of phase space with negative or positive momen‐ tum. One is to use half lines and *t* in the range from -*π* to *π*, with the curve *t* =0 coinciding with the positive *p* axes. The other option is to use the complete curve including positive and negative momentum values and with *t* ∈( - *π* / 2, *π* / 2). In the first option, the positive momentum part of a probability density will correspond to the range *t* ∈( - *π* / 2, *π* / 2), and the negative values will correspond to *t* ∈(-*π*, - *π* 2 ) ∪(*π* / 2, *π*). We take this option.

Now, based on the equalities derived in this chapter, we find the following relationship for a marginal density dependent only upon *H* (*z*), assuming that the function *ρ*(*H* ) can be writ‐ ten as a power series of *H* , *ρ*(*H* )=∑ *ρi H i* ,

*i*

$$\operatorname{e}^{-\tau \mathsf{L}\_H} \rho(H) = \sum\_{n} \frac{(\cdot \cdot \tau)^n}{n!} \mathsf{L}\_H^n \sum\_{i} \rho\_i H^i \stackrel{i}{=} \sum\_{i} \rho\_i H^i = \rho(H) \ , \tag{57}$$

where we have made use of the equality L*H H* =0. Then, a function of *H* does not evolve in time, it is a steady state. For a marginal function dependent upon *t*, we also have that

$$e^{-\tau \mathcal{L}\_H} \rho(t) = e^{\tau d/dt} \rho(t) = \rho(t+\tau) \,. \tag{58}$$

**Figure 2.** Contour plots of the time evolution of a probability density on phase-space and on energy-time space. Initial densities (a) in phase space, and (b) in energy-time space. (d) Evolution in energy-time space is accomplished by a shift

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19

This behaviour is also observed in quantum systems. Time eigenfunctions can be defined in a similar way as for classical systems. We start with a coordinate eigenfunction |*q* > for the

ℏ |*q* =0> . (61)

ℏ |ψ> =*ψ*(q=0;t) , (62)

ℏ |ψ> = <*t* + *τ*|ψ> . (63)

along the *t*axes. (c) In phase space, the density is also translated to the corresponding time eigensurfaces.

*it H* ^

> *it H* ^

*it H* ^ ℏ *eiτ H* ^

Which is the time dependent wave function, in the coordinate representation, and evaluated

Then, time evolution is the translation in time representation, without a change in shape. Note that the variable *τ* is the time variable that appears in the Schrödinger equation for the

eigenvalue *q* =0 and propagate it in time. This will be our time eigenstate

<*t*|*ψ*> = <*q* =0|*e*-

at *q* =0. This function is the time component of the wave function.

The time component of a propagated wave function for a time *τ* is

<*t*|*ψ*(*τ*)> = <*q* =0|*e*-

wave function.

The projection of a wave function onto this vector is

where we have made use of the result that *d dt* = - L*H* . Therefore, a function of *t* is only shifted in time without changing its shape.

For a function of *H* and *t* we find that

$$e^{-\tau \mathcal{L}\_H} \rho(H, t) = e^{-\tau d/dt} \rho(H, t) = \rho(H, t + \tau) \,. \tag{59}$$

This means that evolution in energy-time space also is quite simple, it is only a shift of the function along the *t* axes without a change of shape.

So, let us take a concrete probability density and let us evolve it in time. The probability density, in phase space, that we will consider is

$$\rho(z) = H(z)e^{-\left( (q \cdot q\_0)^2 + (p \cdot p\_0)^2 \right) / 2\sigma^2} \,\, \, \, \, \tag{60}$$

with (*q*0, *p*0) =(1,2) and *σ* =1. A contour plot of this density in phase-space is shown in (a) of Fig. 2. The energy-time components of this density are shown in (b) of the same figure. Time evolution by an amount *τ* correspond to a translation along the *t* axes, from *t* to *t* + *τ*, with‐ out changing the energy values. This translation is illustrated in (d) of Fig. 2 in energy-time space and in (c) of the same figure in phase-space.

Recall that the whole function *ρ*(*z*) is translated in time with the propagator *e*-*τ*L*H* . Then, there are two times involved here, the variable *t* as a coordinate and the shift in time *τ*. The latter is the time variable that appears in the Liouville equation of motion *dρ*(*z*; *τ*) *dτ* = - L*H ρ*(*z*;*τ*).

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables http://dx.doi.org/10.5772/53598 19

**Figure 2.** Contour plots of the time evolution of a probability density on phase-space and on energy-time space. Initial densities (a) in phase space, and (b) in energy-time space. (d) Evolution in energy-time space is accomplished by a shift along the *t*axes. (c) In phase space, the density is also translated to the corresponding time eigensurfaces.

This behaviour is also observed in quantum systems. Time eigenfunctions can be defined in a similar way as for classical systems. We start with a coordinate eigenfunction |*q* > for the eigenvalue *q* =0 and propagate it in time. This will be our time eigenstate

$$\left. \right|\_{t>} = e^{\frac{\hat{\imath}}{\pi}} \left. \right|\_{q=0>} \tag{61}$$

The projection of a wave function onto this vector is

and negative momentum values and with *t* ∈( - *π* / 2, *π* / 2). In the first option, the positive momentum part of a probability density will correspond to the range *t* ∈( - *π* / 2, *π* / 2), and

Now, based on the equalities derived in this chapter, we find the following relationship for a marginal density dependent only upon *H* (*z*), assuming that the function *ρ*(*H* ) can be writ‐

where we have made use of the equality L*H H* =0. Then, a function of *H* does not evolve in

This means that evolution in energy-time space also is quite simple, it is only a shift of the

So, let us take a concrete probability density and let us evolve it in time. The probability

with (*q*0, *p*0) =(1,2) and *σ* =1. A contour plot of this density in phase-space is shown in (a) of Fig. 2. The energy-time components of this density are shown in (b) of the same figure. Time evolution by an amount *τ* correspond to a translation along the *t* axes, from *t* to *t* + *τ*, with‐ out changing the energy values. This translation is illustrated in (d) of Fig. 2 in energy-time

Recall that the whole function *ρ*(*z*) is translated in time with the propagator *e*-*τ*L*H* . Then, there are two times involved here, the variable *t* as a coordinate and the shift in time *τ*. The latter is the time variable that appears in the Liouville equation of motion

time, it is a steady state. For a marginal function dependent upon *t*, we also have that

2 ) ∪(*π* / 2, *π*). We take this option.

*ρ*(*t*)=*ρ*(*t* + *τ*) . (58)

*dt* = - L*H* . Therefore, a function of *t* is only shifted

*ρ*(*H* , *t*)=*ρ*(*H* , *t* + *τ*) . (59)

, (60)

=*ρ*(*H* ) , (57)

the negative values will correspond to *t* ∈(-*π*, - *π*

*e*-*τ*L*H ρ*(*H* )=∑

where we have made use of the result that *d*

in time without changing its shape.

For a function of *H* and *t* we find that

*i ρi H i* ,

( - *τ*)*n n* ! L*H n* ∑ *i ρi H i* =∑ *i ρi H i*

*e*-*τ*L*H ρ*(*t*)=*e τd* /*dt*

*n*

*e*-*τ*L*H ρ*(*H* , *t*)=*e*-*τd* /*dt*

*ρ*(*z*)=*H* (*z*)*e*

function along the *t* axes without a change of shape.

density, in phase space, that we will consider is

space and in (c) of the same figure in phase-space.

*dρ*(*z*; *τ*)

*dτ* = - L*H ρ*(*z*;*τ*).

ten as a power series of *H* , *ρ*(*H* )=∑

18 Advances in Quantum Mechanics

$$<\psi \mid \psi> =  = \psi \{ \mathbf{q}=0; \mathbf{t} \} \tag{62}$$

Which is the time dependent wave function, in the coordinate representation, and evaluated at *q* =0. This function is the time component of the wave function.

The time component of a propagated wave function for a time *τ* is

$$<\langle t \mid \psi(\tau) > =  =  . \tag{63}$$

Then, time evolution is the translation in time representation, without a change in shape. Note that the variable *τ* is the time variable that appears in the Schrödinger equation for the wave function.

Now, assuming a discrete energy spectrum with energy eigenvalue *En* and corresponding eigenfunction |*n* >, in the energy representation we have that

$$<\mu \mid \psi(\tau) > = <\nu \mid e^{\cdot \frac{\hat{\imath}H}{\hbar}} \Big| \psi > = e^{\cdot \cdot \frac{\hat{\imath}H\_n}{\hbar}} <\nu \Big| \psi > \Big. \tag{64}$$

[5] Jaffé C. Classical Liouville mechanics and intramolecular relaxation dynamics. The

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

http://dx.doi.org/10.5772/53598

21

[6] Jaffé C and Brumer C. Classical-quantum correspondence in the distribution dynam‐

[7] Jaffé C. Semiclassical quantization of the Liouville formulation of classical mechanics.

[8] Jaffé C. Sheldon Kanfer and Paul Brumer, Classical analog of pure-state quantum dy‐

ics of integrable systems. Journal of Chemical Physics 1985; 82 2330.

Journal of Physical Chemistry 1984; 88 4829.

Journal of Chemical Physics 1988; 88 7603.

namics. Physical Review Letters 1985; 54 8.

[10] Galapon EA, Proc R Soc Lond A 2002; 458 451

[11] Galapon EA, Proc R Soc Lond A 2002; 458 2671

[14] Garrison JC and Wong J, J Math Phys 1970; 11 2242

[15] Torres-Vega G, J Phys A: Math Theor 45, 215302 (2012)

[12] Galapon EA, quant-ph/0303106

[13] Galindo A, Lett Math Phys 1984; 8 495

[9] Pauli W. Handbuch der Physics. Berlin: Springer-Verlag; 1926

i.e. the wave function in energy space only changes its phase after evolution for a time *τ*.

### **4. Concluding remarks**

Once that we have made use of the same concepts in both classical and quantum mechanics, it is more easy to understand quantum theory since many objects then are present in both theories.

Actually, there are many things in common for both classical and quantum systems, as is the case of the eigensurfaces and the eigenfunctions of conjugate variables, which can be used as coordinates for representing dynamical quantities.

Another benefit of knowing the influence of conjugate dynamical variables on themselves and of using the same language for both theories lies in that some puzzling things that are found in one of the theories can be analysed in the other and this helps in the understanding of the original puzzle. This is the case of the Pauli theorem [9-14] that prevents the existence of a hermitian time operator in Quantum Mechanics. The classical analogue of this puzzle is found in Reference [15].

These were some of the properties and their consequences in which both conjugate variables participate, influencing each other.

### **Author details**

Gabino Torres-Vega

Physics Department, Cinvestav, México

### **References**

Now, assuming a discrete energy spectrum with energy eigenvalue *En* and corresponding

ℏ |ψ> =*e*-

Once that we have made use of the same concepts in both classical and quantum mechanics, it is more easy to understand quantum theory since many objects then are present in both

Actually, there are many things in common for both classical and quantum systems, as is the case of the eigensurfaces and the eigenfunctions of conjugate variables, which can be used as

Another benefit of knowing the influence of conjugate dynamical variables on themselves and of using the same language for both theories lies in that some puzzling things that are found in one of the theories can be analysed in the other and this helps in the understanding of the original puzzle. This is the case of the Pauli theorem [9-14] that prevents the existence of a hermitian time operator in Quantum Mechanics. The classical analogue of this puzzle is

These were some of the properties and their consequences in which both conjugate variables

[1] Woodhouse NMJ. Geometric Quantization. Oxford: Osford University Press; 1991.

[4] Torres-Vega G, Theoretical concepts of quantum mechanics. Rijeka: InTech; 2012.

i.e. the wave function in energy space only changes its phase after evolution for a time *τ*.

*iτ*En

ℏ <*n*|ψ> , (64)

*iτ H* ^

eigenfunction |*n* >, in the energy representation we have that

<*n*|*ψ*(*τ*)> = <*n*|*e*-

coordinates for representing dynamical quantities.

**4. Concluding remarks**

20 Advances in Quantum Mechanics

found in Reference [15].

**Author details**

Gabino Torres-Vega

**References**

participate, influencing each other.

Physics Department, Cinvestav, México

[2] Wigner E. Phys Rev A 1932; 40 749

[3] Husimi K. Proc Phys Math Soc Jpn 1940; 22 264

theories.

**Chapter 2**

**Provisional chapter**

**Classical and Quantum Correspondence in Anisotropic**

If the classical behavior of a given quantum system is chaotic, how is it reflected in the quantum properties of the system? To elucidate this correspondence is the main theme of the quantum chaos study. With the advent of nanophysics techniques, this has become also of experimental importance. With the advent of new technology, various quantum systems are now challenging us. These include nano-scale devices, laser trapping of atoms, the Bose-Einstein condensate, Rydberg atoms, and even web of chaos is observed in superlattices. In this note we devote ourselves to the investigation of the quantum scars which occurs in the Anisotropic Kepler Problem (AKP) – the classical and quantum physics of an electron trapped around a proton in semiconductors. The merit of AKP is that its chaotic property can be controlled by changing the anisotropy from integrable Kepler limit down ergodic limit where the tori are completely collapsed and isolated unstable periodic orbits occupy the classical phase space. Thus in AKP we are able to investigate the classical quantum correspondence at varying chaoticity. Furthermore each unstable periodic orbit (PO) can be coded in a Bernoulli code which is a large merit in the formulation of quantum chaos in term

The AKP is an old home ground of the quantum chaos study. Its low energy levels were used as a test of the periodic orbit theory in the seminal work of Gutzwiller [3-7]. Then an efficient matrix diagonalization scheme was devised by Wintgen et al. (WMB method) [8]. With this method, the statistics of up to nearly 8000 AKP quantum levels were examined and it was found that the quantum level statistics of AKP change from Poisson to Wigner distribution with the increase of mass anisotropy [9]. Furthermore, an intriguing classical Poincaré surface of section (POS) was found at medium anisotropy (*γ* ≡ *m*(light)/*m*(heavy) = 0.8),

> ©2012 Shimada et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Sumiya et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 Sumiya et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

**Classical and Quantum Correspondence in**

**Kepler Problem**

http://dx.doi.org/10.5772/55208

10.5772/55208

**1. Introduction**

Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada

of the periodic orbit theory (POT) [1, 2, 6].

Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada

**Anisotropic Kepler Problem**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Provisional chapter**

### **Classical and Quantum Correspondence in Anisotropic Kepler Problem Classical and Quantum Correspondence in Anisotropic Kepler Problem**

Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55208 10.5772/55208

### **1. Introduction**

If the classical behavior of a given quantum system is chaotic, how is it reflected in the quantum properties of the system? To elucidate this correspondence is the main theme of the quantum chaos study. With the advent of nanophysics techniques, this has become also of experimental importance. With the advent of new technology, various quantum systems are now challenging us. These include nano-scale devices, laser trapping of atoms, the Bose-Einstein condensate, Rydberg atoms, and even web of chaos is observed in superlattices.

In this note we devote ourselves to the investigation of the quantum scars which occurs in the Anisotropic Kepler Problem (AKP) – the classical and quantum physics of an electron trapped around a proton in semiconductors. The merit of AKP is that its chaotic property can be controlled by changing the anisotropy from integrable Kepler limit down ergodic limit where the tori are completely collapsed and isolated unstable periodic orbits occupy the classical phase space. Thus in AKP we are able to investigate the classical quantum correspondence at varying chaoticity. Furthermore each unstable periodic orbit (PO) can be coded in a Bernoulli code which is a large merit in the formulation of quantum chaos in term of the periodic orbit theory (POT) [1, 2, 6].

The AKP is an old home ground of the quantum chaos study. Its low energy levels were used as a test of the periodic orbit theory in the seminal work of Gutzwiller [3-7]. Then an efficient matrix diagonalization scheme was devised by Wintgen et al. (WMB method) [8]. With this method, the statistics of up to nearly 8000 AKP quantum levels were examined and it was found that the quantum level statistics of AKP change from Poisson to Wigner distribution with the increase of mass anisotropy [9]. Furthermore, an intriguing classical Poincaré surface of section (POS) was found at medium anisotropy (*γ* ≡ *m*(light)/*m*(heavy) = 0.8),

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Sumiya et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Sumiya et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Shimada et al., licensee InTech. This is an open access chapter distributed under the terms of the

which indicates remnants of tori (cantori) in the classical phase space [9]. Thus, over two decades from the early 70th, AKP was a good testing ground of theories (along with billiards) as well as a constant source of important information to quantum chaos studies. However, there has not been much recent theory investigation on AKP. Especially, to our knowledge, the quantum scar of the classical periodic orbits in AKP has not been directly examined, even though intriguing phenomena was discovered by Heller [10] in 1984. On the other hand, for an analogous system – the hydrogen under a magnetic field (diamagnetic Kepler problem (DKP)), the scars of periodic orbits were extensively studied using highly efficient tool called as scar strength functions [11]. We note that AKP is by far simpler; for DKP it is necessary to code the POs by a sequence of symbols consisting of three letters.

10.5772/55208

25

http://dx.doi.org/10.5772/55208

seemingly contradicting features intriguingly compromise. The localization patterns in the wave functions or Husimi functions are swapped between two eigenstates of energy at every avoiding crossing. Repeating successively this swap process characteristic scarring patterns follow the POs responsible to them. In this sense the quantum scarring phenomena are

We first explain how we have prepared the energy levels and wave functions. Then we introduce the indispensable ingredients to study the scars in AKP. After briefly explaining Husimi functions, we explain periodic orbit theory. The quantum scars will be observed

robust. We conclude in section 4.

**2.1. Matrix diagonalization**

*2.1.1. AKP Hamiltonian*

with *γ* = *ν*/*µ* = 1/*µ*2 or

with literature.

*2.1.2. Matrix diagonalization in Sturmian basis*

Firstly, in the Sturmian basis

**2. Manifestation of Scars in AKP**

along the classical unstable periodic orbits.

The AKP Hamiltonian in the dimensionless form is given by

*HG* = 1

or equivalently it may be also written as [9,11] (Harmonic basis)

*H*88 *W* = 1 2 (*p*2 *x* + *p*2

> *H*87 *W* = *p*2

We here summarize WMB method for efficient matrix diagonalization.

2*µ p*2 *x* + 1 2*ν* (*p*2 *y* + *p*2

where *r* = *x*2 + *y*2 + *z*2 and *µ* > *ν*[1,4] with which POT was formulated in the history,

*x* + *p*2

as used in WMB (Sturmian basis) [8]. We recapitulate POT predictions in terms of (1), our formula for AKP eigenvalue calculation in tensored harmonic wave function basis in terms of (2), and we discuss quantum scars using energy values in (3) in order to facilitate comparison

*y*) + *γ* 2 *p*2 *z* − 1

*y* + *γp*2

*z* − 2

*z* ) − 1

Classical and Quantum Correspondence in Anisotropic Kepler Problem

*r* (1)

*r* (2)

*r* (3)

Recently the level statistics of AKP was examined from the random matrix theory view [12]. It was considered that the AKP level statistic in the transitive region from Poisson to Wigner distribution correspond to the critical level statistics of an extended GOE random matrix theory and it was conjectured that the wave functions should exhibit characteristic multifractality. This aspect has been further developed in [13, 14]; it is considered that *Anderson transition* occurs in the quantum physics of a class of physical systems such as AKP and periodically driven kicked rotator in their critical parameter regions. Further very recently a well devised new solid state experiment has been conducted for AKP and ADKP [15, 16]. We also refer [17] for a recent overview including this interesting conjecture.

Such is the case we have recently conducted AKP high accuracy matrix diagonalization based on the WMB method. This is not a perturbation calculation; the anisotropy term is not regarded as a perturbation and the full Hamiltonian matrix is diagonalized. Thus the approximation comes only from the size of the matrix. But, as a trade-off, a scaling parameter is unavoidably included; it is crucial to choose a correct parameter value at every anisotropy parameter. We have derived a simple rule of thumb to choose a suitable value [17]. After comparing with original WMB result in Sturmian basis, we have also worked with tensored-harmonic-wavefunction basis (THWFB) [11], which is more suitable for the Husimi function calculation to investigate the quantum scars. Our contribution here is the calculation of anisotropy term in the AKP Hamiltonian in THWFB [17], which is harder than the diamagnetic case. Comparing the results from two independent bases we have verified that both results agree completely thus the choices of scaling parameters (in both bases) are validated.

Aimed by these numerical data, we report in section 2 salient evidences of quantum scars in AKP for the first time. We compare the features of various known observables; thus this section will serve as a comparative test of methods and fulfills the gap in the literature pointed out above. Most interesting is the test using the scar strength function. We show that even in the ergodic regime (*γ* = 0.2), we can quantitatively observe that prominent periodic orbits systematically contribute to the quantum theory endowed with random energy spectrum.

In section 3 we investigate that how the scaring phenomena are affected by the variation of the anisotropy parameter. It is well known that the energy levels show successive avoiding crossings. On the other hand, in the periodic orbit formula, each term in the series for the density of states (DOS) consists of a contribution of an unstable PO with a pole (with an imaginary part given by the Lyapunov exponent of the PO) at the Bohr-Sommerfeld-type energy; thus each term smoothly varies with the anisotropy. We show that how these two seemingly contradicting features intriguingly compromise. The localization patterns in the wave functions or Husimi functions are swapped between two eigenstates of energy at every avoiding crossing. Repeating successively this swap process characteristic scarring patterns follow the POs responsible to them. In this sense the quantum scarring phenomena are robust. We conclude in section 4.

### **2. Manifestation of Scars in AKP**

We first explain how we have prepared the energy levels and wave functions. Then we introduce the indispensable ingredients to study the scars in AKP. After briefly explaining Husimi functions, we explain periodic orbit theory. The quantum scars will be observed along the classical unstable periodic orbits.

#### **2.1. Matrix diagonalization**

#### *2.1.1. AKP Hamiltonian*

2 Advances in Quantum Mechanics

validated.

energy spectrum.

which indicates remnants of tori (cantori) in the classical phase space [9]. Thus, over two decades from the early 70th, AKP was a good testing ground of theories (along with billiards) as well as a constant source of important information to quantum chaos studies. However, there has not been much recent theory investigation on AKP. Especially, to our knowledge, the quantum scar of the classical periodic orbits in AKP has not been directly examined, even though intriguing phenomena was discovered by Heller [10] in 1984. On the other hand, for an analogous system – the hydrogen under a magnetic field (diamagnetic Kepler problem (DKP)), the scars of periodic orbits were extensively studied using highly efficient tool called as scar strength functions [11]. We note that AKP is by far simpler; for DKP it is necessary to

Recently the level statistics of AKP was examined from the random matrix theory view [12]. It was considered that the AKP level statistic in the transitive region from Poisson to Wigner distribution correspond to the critical level statistics of an extended GOE random matrix theory and it was conjectured that the wave functions should exhibit characteristic multifractality. This aspect has been further developed in [13, 14]; it is considered that *Anderson transition* occurs in the quantum physics of a class of physical systems such as AKP and periodically driven kicked rotator in their critical parameter regions. Further very recently a well devised new solid state experiment has been conducted for AKP and ADKP [15, 16]. We also refer [17] for a recent overview including this interesting conjecture.

Such is the case we have recently conducted AKP high accuracy matrix diagonalization based on the WMB method. This is not a perturbation calculation; the anisotropy term is not regarded as a perturbation and the full Hamiltonian matrix is diagonalized. Thus the approximation comes only from the size of the matrix. But, as a trade-off, a scaling parameter is unavoidably included; it is crucial to choose a correct parameter value at every anisotropy parameter. We have derived a simple rule of thumb to choose a suitable value [17]. After comparing with original WMB result in Sturmian basis, we have also worked with tensored-harmonic-wavefunction basis (THWFB) [11], which is more suitable for the Husimi function calculation to investigate the quantum scars. Our contribution here is the calculation of anisotropy term in the AKP Hamiltonian in THWFB [17], which is harder than the diamagnetic case. Comparing the results from two independent bases we have verified that both results agree completely thus the choices of scaling parameters (in both bases) are

Aimed by these numerical data, we report in section 2 salient evidences of quantum scars in AKP for the first time. We compare the features of various known observables; thus this section will serve as a comparative test of methods and fulfills the gap in the literature pointed out above. Most interesting is the test using the scar strength function. We show that even in the ergodic regime (*γ* = 0.2), we can quantitatively observe that prominent periodic orbits systematically contribute to the quantum theory endowed with random

In section 3 we investigate that how the scaring phenomena are affected by the variation of the anisotropy parameter. It is well known that the energy levels show successive avoiding crossings. On the other hand, in the periodic orbit formula, each term in the series for the density of states (DOS) consists of a contribution of an unstable PO with a pole (with an imaginary part given by the Lyapunov exponent of the PO) at the Bohr-Sommerfeld-type energy; thus each term smoothly varies with the anisotropy. We show that how these two

code the POs by a sequence of symbols consisting of three letters.

The AKP Hamiltonian in the dimensionless form is given by

$$H\_G = \frac{1}{2\mu}p\_x^2 + \frac{1}{2\nu}(p\_y^2 + p\_z^2) - \frac{1}{r} \tag{1}$$

where *r* = *x*2 + *y*2 + *z*2 and *µ* > *ν*[1,4] with which POT was formulated in the history, or equivalently it may be also written as [9,11] (Harmonic basis)

$$H\_W^{88} = \frac{1}{2}(p\_x^2 + p\_y^2) + \frac{\gamma}{2}p\_z^2 - \frac{1}{r} \tag{2}$$

with *γ* = *ν*/*µ* = 1/*µ*2 or

$$H\_W^{87} = p\_x^2 + p\_y^2 + \gamma p\_z^2 - \frac{2}{r} \tag{3}$$

as used in WMB (Sturmian basis) [8]. We recapitulate POT predictions in terms of (1), our formula for AKP eigenvalue calculation in tensored harmonic wave function basis in terms of (2), and we discuss quantum scars using energy values in (3) in order to facilitate comparison with literature.

#### *2.1.2. Matrix diagonalization in Sturmian basis*

We here summarize WMB method for efficient matrix diagonalization.

Firstly, in the Sturmian basis

$$
\langle \overrightarrow{r} \vert n\ell m \rangle = \frac{1}{r} \sqrt{\frac{n!}{(2\ell+n+1)!}} e^{-\frac{\lambda r}{2}} (\lambda r)^{\ell+1} L\_n^{2\ell+1}(\lambda r) Y\_{\ell m}(\theta, \varphi) \tag{4}
$$

27

|**Ψ**� = *E* |**Ψ**�. (11)

http://dx.doi.org/10.5772/55208

and the AKP Schrödinger equation becomes

 ∆(2) *µ* + ∆(2) *ν* + 1 − *γ* 2

�*µ*, *ν* | *i*, *j*, *κ*� = *κ*

*π*

with a scaling parameter *κ* and, corresponding to *ε* in (6), we introduce a parameter

*En* = −*κ*2 *n* 2 *ε*˜ = − 2 Λ2 *n*

*ε*˜

We have found precise agreement between our calculations by the Sturmian basis and by the harmonic oscillator basis which in turn validates our choices of scaling parameter *ε* and *ε*˜. For the calculation of Husimi functions and scar strength function which uses Husimi functions, we use the THWFB since the projection of the basis functions to the Gaussian

*ε*˜ = 2

and we solve (12) after transforming it into the matrix equation of WMB form with eigenvalues Λ*n* = 2/*κn*. The matrix element calculation of the mass anisotropy term in (12) is somewhat involved and we refer to [17] for detail. Energy levels are then determined

*∂*2 *∂z*2 − 2

> *µ*2 + *ν*2

Multiplying by *µ*2 + *ν*2and swapping the Coulombic interaction term and the *E* term one

Thanks to the semi-parabolic coordinates, the Coulombic singularity has removed [19] for *γ* = 1. Corresponding to the Sturmian basis with a scaling parameter *λ* in (4), we introduce

*Li*(*κµ*2)*Lj*(*κν*2) exp

*µ*2 + *ν*2

Classical and Quantum Correspondence in Anisotropic Kepler Problem

 *∂*2 *∂z*2

−*µ*2 + *ν*2 2

*κ*2 (13)

*ε*˜ (14)

∗ ≈ *γ* (15)

 − 1 2 (*µ*2 + *ν*2)

 −1 2 ∆(2) *µ* + ∆(2) *ν* + |*E*| *µ*2 + *ν*2 + 1 − *γ* 2

the harmonic wave function basis

The best value of *ε*˜ is given by

packets are easy to calculate [17].

which is similar to (7).

obtains

by

with a scaling parameter *λ*, the Schrödinger equation of the AKP becomes a matrix equation:

$$\left[ -\lambda \overleftarrow{\Delta^{(3)}} + (1 - \gamma)\lambda \overleftarrow{\frac{\partial^2}{\partial z^2}} - 2\frac{\leftrightarrow}{r} \right] \mathbf{v} = \frac{E}{\lambda} \overleftarrow{\text{Id}} \,\mathbf{v} \tag{5}$$

Dividing the whole equation by *λ* and packing *E*/*λ*2 into a parameter *ε*, one obtains

$$\overset{\textstyle \longleftrightarrow}{\widetilde{M}} \mathbf{Y} \equiv \left[ \overset{\textstyle \longleftrightarrow}{-\Delta^{\langle 3 \rangle}} + (1 - \gamma) \frac{\overset{\textstyle \longleftrightarrow}{\partial^2}}{\partial z^2} - \epsilon \overset{\textstyle \longleftrightarrow}{\operatorname{Id}} \right] \mathbf{Y} = \frac{2}{\lambda} \mathbf{Y}. \tag{6}$$

This *ǫ* is to be fixed at some constant value. In principle any value will do, but for finite size of Hamiltonian matrix, the best choice is given [17] approximately

$$
\varepsilon^\* \simeq -\frac{1}{4}\gamma.\tag{7}
$$

With this choice, we can get the largest number of reliable energy levels at a given matrix size. The ratio of reliable levels to the matrix size can be estimated as

$$R\_{eff} \simeq \sqrt{\gamma} \,. \tag{8}$$

After fixing *ε*, the diagonalization of (6) is performed for 2/*λi* s and finally we obtain the energy eigenvalues by

$$E\_{\hat{l}} = \varepsilon \lambda\_{\hat{l}}^2. \tag{9}$$

#### *2.1.3. Matrix diagonalization in Sturmian basis*

For the (tensored) harmonic wave function basis (THWFB) [11] we convert the Hamiltonian of AKP into the Hamiltonian of two of two-dimensional harmonic oscillators.

For this purpose semi-parabolic coordinates are introduced

$$
\mu \nu = \rho = \sqrt{x^2 + y^2}, \quad \frac{1}{2}(\mu^2 - \nu^2) = z, \quad \phi = \tan^{-1}\left(\frac{y}{x}\right) \tag{10}
$$

and the AKP Schrödinger equation becomes

4 Advances in Quantum Mechanics

energy eigenvalues by

*2.1.3. Matrix diagonalization in Sturmian basis*

*µν* = *ρ* =

��*r*|*n*ℓ*m*� = 1

 −*λ*

←→*M* **Ψ** ≡

*r* �

←→

 −

←→

of Hamiltonian matrix, the best choice is given [17] approximately

size. The ratio of reliable levels to the matrix size can be estimated as

*n*! (2ℓ + *n* + 1)!

∆(3) + (1 − *γ*)*λ*

*e* − *λr*

with a scaling parameter *λ*, the Schrödinger equation of the AKP becomes a matrix equation:

←→ *∂*2 *∂z*2 − 2

> ←→ *∂*2 *∂z*2 − *ǫ*

This *ǫ* is to be fixed at some constant value. In principle any value will do, but for finite size

With this choice, we can get the largest number of reliable energy levels at a given matrix

After fixing *ε*, the diagonalization of (6) is performed for 2/*λi* s and finally we obtain the

*Ei* = *ελ*2

For the (tensored) harmonic wave function basis (THWFB) [11] we convert the Hamiltonian

(*µ*2 − *ν*2) = *z*, *φ* = tan−1

of AKP into the Hamiltonian of two of two-dimensional harmonic oscillators.

*x*2 + *y*2, 1

2

For this purpose semi-parabolic coordinates are introduced

�

Dividing the whole equation by *λ* and packing *E*/*λ*2 into a parameter *ε*, one obtains

∆(3) + (1 − *γ*)

*ε* ∗ ≃ −1 4 ←→1 *r*

 **Ψ** = *E λ*

←→Id **Ψ** = 2

2 (*λr*)ℓ+1*L*2ℓ+1 *n* (*λr*)*Y*ℓ*m*(*θ*, *ϕ*) (4)

Id **Ψ** (5)

*λ* **Ψ**. (6)

*γ*. (7)

*i* . (9)

� *y x* �

(10)

*Reff* ≃ √*γ*. (8)

←→

$$\left[ -\frac{1}{2\left(\mu^2 + \nu^2\right)} \left( \Delta\_{\mu}^{(2)} + \Delta\_{\nu}^{(2)} \right) + \frac{1-\gamma}{2} \frac{\partial^2}{\partial z^2} - \frac{2}{\mu^2 + \nu^2} \right] |\mathbf{Y}\rangle = E |\mathbf{Y}\rangle\_{\cdot} \tag{11}$$

Multiplying by *µ*2 + *ν*2and swapping the Coulombic interaction term and the *E* term one obtains

$$\left[-\frac{1}{2}\left(\Delta\_{\mu}^{(2)} + \Delta\_{\nu}^{(2)}\right) + |E|\left(\mu^2 + \nu^2\right) + \frac{1-\gamma}{2}\left(\mu^2 + \nu^2\right)\frac{\partial^2}{\partial z^2}\right]|\Psi\rangle = 2\left|\Psi\right\rangle. \tag{12}$$

Thanks to the semi-parabolic coordinates, the Coulombic singularity has removed [19] for *γ* = 1. Corresponding to the Sturmian basis with a scaling parameter *λ* in (4), we introduce the harmonic wave function basis

$$\langle \mu, \nu \mid i, j, \kappa \rangle = \frac{\kappa}{\pi} L\_i(\kappa \mu^2) L\_j(\kappa \nu^2) \exp\left(-\frac{\mu^2 + \nu^2}{2}\right).$$

with a scaling parameter *κ* and, corresponding to *ε* in (6), we introduce a parameter

$$
\tilde{\varepsilon} = 2 \frac{|E|}{\kappa^2} \tag{13}
$$

and we solve (12) after transforming it into the matrix equation of WMB form with eigenvalues Λ*n* = 2/*κn*. The matrix element calculation of the mass anisotropy term in (12) is somewhat involved and we refer to [17] for detail. Energy levels are then determined by

$$E\_n = -\frac{\kappa\_n^2}{2}\tilde{\varepsilon} = -\frac{2}{\Lambda\_n^2}\tilde{\varepsilon}\tag{14}$$

The best value of *ε*˜ is given by

$$
\mathfrak{E}^\* \approx \gamma \tag{15}
$$

which is similar to (7).

We have found precise agreement between our calculations by the Sturmian basis and by the harmonic oscillator basis which in turn validates our choices of scaling parameter *ε* and *ε*˜.

For the calculation of Husimi functions and scar strength function which uses Husimi functions, we use the THWFB since the projection of the basis functions to the Gaussian packets are easy to calculate [17].

#### **2.2. Husimi function**

Husimi function is defined via the scalar product of the wave function |*ψ*� with a coherent state(CHS) |*q*0, *p*0� of the system [11]:

$$\mathcal{W}^{Hus}\_{\psi} \left( q\_{0\prime} \ p\_0 \right) = \left| \langle \psi \mid q\_{0\prime} \ p\_0 \rangle \right|^2 \tag{16}$$

10.5772/55208

29

(21)

(24)

(20)

http://dx.doi.org/10.5772/55208

*ρ*(*E*) ≡ ∑

DOS

Because

DOS

(24) is given by

*ρ*(*E*) ≈ − 1

*π* Im

the stationary phase condition gives

*n*

*dq*′ *G*˜(*q*′′, *q*′

, *E*)

*p*′′ = *∂S*(*q*′′, *q*′

0 = *∂S*(*q*′

*r*∈*PO*

and the matrix *M* stands for the monodromy matrix of the primary PO.

2 [det(*Mn* − 1)]

*ρ*(*E*) ≃ *ρ*(*E*) + Im ∑

1

*δ*(*E* − *En*) = − 1

*π* Im *Trn*

where in the second equality an identity 1/(*x* + *iε*) = *P*(1/*x*) − *iπδ*(*x*) is used and trace is taken over all energy eigenstates {|*n*�}. Trading this tracing with the tracing over the eigenstates of coordinate operator {|*q*�}, we obtain a semiclassical approximation for the

*q*′′=*q*′ = − 1

)

, *q*′ )

which dictates the periodic orbits. We obtain finally the periodic orbit theory formula for the

*Tr πh*¯ ∑ *n*�=0

Here the first sum runs over all primitive POs and the *n* sum counts the repetitions of each peridic orbit; *Tr*, *Sr*, and *lr* denote the period, action, and Maslov index of the primary PO,

In AKP *m* = 0 sector, the motion is restricted in a fixed plane which includes the heavy axis, and the problem essentially reduces to two dimensional one. (Later on the three dimensional feature is recovered only by the proper choice of the Maslov index [4]). As for AKP unstable periodic orbits, *M* has two eigenvalues *eu* and *e*−*u* (hyperbolic case) and the determinant in

1

exp *in Sr h*¯ − *π* 2 *lr*

[det((*Mr*)*n* − 1)]

*π* Im

*∂q*′′ , *p*′ = −*∂S*(*q*′′, *q*′

The integration over *q*′ can be again approximated by a stationary phase approximation.

 1 *E* + *iε* − *H*ˆ

Classical and Quantum Correspondence in Anisotropic Kepler Problem

*dq*′ ∑ Γ

*A*Γ exp *i*

*∂q*′ = *p*′′ − *p*′ (23)

1 2

2 = −*i* sinh (*nu*/2) (25)

)

*h*¯ *S*Γ − *i ν*Γ 2 .

*∂q*′ (22)

A detailed account is given in [17].

#### **2.3. Periodic orbit theory**

#### *2.3.1. Periodic orbit theory and the density of state*

Let us recapitulate Gutzwiller's periodic orbit theory [4,20]. The starting point is Feynman's path integral formula for the propagator of a particle from *q*′ to *q*′′ during the time interval 0 to *T*;

$$\mathbb{K}\left(q'',q',T\right) \equiv \left\langle q'' \middle| \exp\left(-i\frac{H}{\hbar}T\right) \left| q' \right\rangle = \int\_{q'}^{q''} D\left[q\right] e^{\frac{i}{\hbar}\int\_{0}^{T} L(q,q,t)dt} \tag{17}$$

The Green function (response function) is given by the Fourier transformation of the propagator

$$G\left(q^{\prime\prime}, q^{\prime}, E\right) \equiv -\frac{i}{\hbar} \int\_0^\infty dt e^{\frac{iEt}{\hbar}} K\left(q^{\prime\prime}, q^{\prime}, T\right).$$

where *E* has infinitesimally small imaginary part for convergence. Thus we have

$$\mathcal{G}\left(q'',q',E\right) = \left\langle q'' \middle| \begin{array}{c} 1 \\ E+i\varepsilon-\hat{H} \end{array} \middle| q' \right\rangle = -\frac{i}{\hbar} \int\_0^\infty dt e^{\frac{iEt}{\hbar}} \left[ \int\_{q'}^{q''} D\left[q\right] e^{\frac{i}{\hbar} \int\_0^T L(q,q,t)dt} \right] \tag{18}$$

By a stationary approximation we obtain a semiclassical formula for the Green function

$$\tilde{G}\left(q'',q',E\right) \simeq \sum\_{\Gamma} A\_{\Gamma} \exp\left(\frac{i}{\hbar}\mathcal{S}\_{\Gamma} - i\frac{\nu\_{\Gamma}}{2}\right) \tag{19}$$

where Γ denotes a classically arrowed orbit, *ν*Γ is the number of conjugate points on the orbit, and the amplitude *A*Γ accounts for the Van Vleck determinant. Note that the principal function in (18) is changed into the action *S* = *q*′′ *q*′ *pdq* and the phase *iπ*/4 from the stationary point approximation is shifted into *A*Γ.

Now the density of states is given by

$$\rho(E) \equiv \sum\_{\mathfrak{n}} \delta(E - E\_{\mathfrak{n}}) = -\frac{1}{\pi} \text{Im} \left( T\_{r\_{\mathfrak{n}}} \left( \frac{1}{E + i\varepsilon - \hat{H}} \right) \right) \tag{20}$$

where in the second equality an identity 1/(*x* + *iε*) = *P*(1/*x*) − *iπδ*(*x*) is used and trace is taken over all energy eigenstates {|*n*�}. Trading this tracing with the tracing over the eigenstates of coordinate operator {|*q*�}, we obtain a semiclassical approximation for the DOS

$$\rho(\mathbf{E}) \approx -\frac{1}{\pi} \text{Im} \int d\boldsymbol{q}' \,\,\tilde{\mathbf{G}}(\boldsymbol{q}'', \boldsymbol{q}', \mathbf{E})\big|\_{\boldsymbol{q}'' = \boldsymbol{q}'} = -\frac{1}{\pi} \text{Im} \int d\boldsymbol{q}' \sum\_{\Gamma} A\_{\Gamma} \,\exp\left(\frac{i}{\hbar} \mathbf{S}\_{\Gamma} - i\frac{\nu\_{\Gamma}}{2}\right) \tag{21}$$

The integration over *q*′ can be again approximated by a stationary phase approximation. Because

$$p^{\prime\prime} = \frac{\partial S(q^{\prime\prime}, q^{\prime})}{\partial q^{\prime\prime}}, \ p^{\prime} = -\frac{\partial S(q^{\prime\prime}, q^{\prime})}{\partial q^{\prime}} \tag{22}$$

the stationary phase condition gives

6 Advances in Quantum Mechanics

**2.2. Husimi function**

state(CHS) |*q*0, *p*0� of the system [11]:

A detailed account is given in [17].

*2.3.1. Periodic orbit theory and the density of state*

*G q*′′, *q*′ , *E* ≡ − *i h*¯ ∞ 0

1 *E* + *iε* − *H*ˆ

*G*˜ *q*′′, *q*′ , *E* ≃ ∑ Γ

function in (18) is changed into the action *S* = *q*′′

point approximation is shifted into *A*Γ. Now the density of states is given by

**2.3. Periodic orbit theory**

*K q*′′, *q*′ , *T* ≡ *q*′′ exp −*i H h*¯ *T q*′ = *q*′′

0 to *T*;

propagator

*G q*′′, *q*′ , *E* = *q*′′

Husimi function is defined via the scalar product of the wave function |*ψ*� with a coherent

*ψ* (*q*0, *p*0) = |�*ψ* | *q*0, *p*0�|2

Let us recapitulate Gutzwiller's periodic orbit theory [4,20]. The starting point is Feynman's path integral formula for the propagator of a particle from *q*′ to *q*′′ during the time interval

The Green function (response function) is given by the Fourier transformation of the

where *E* has infinitesimally small imaginary part for convergence. Thus we have

 *q*′ = − *i h*¯ ∞ 0

By a stationary approximation we obtain a semiclassical formula for the Green function

*A*Γ exp(

where Γ denotes a classically arrowed orbit, *ν*Γ is the number of conjugate points on the orbit, and the amplitude *A*Γ accounts for the Van Vleck determinant. Note that the principal

*i h*¯ *S*Γ − *i ν*Γ

*dteiEt h*¯ *K*

*q*′ *D* [*q*]*e*

*q*′′, *q*′ , *T*

*dteiEt h*¯ *q*′′

*i h*¯ *T*

*q*′ *D* [*q*]*e*

*i h*¯ *T* 0 *L*(*q*, *q*˙, *t*)*dt*

*q*′ *pdq* and the phase *iπ*/4 from the stationary

2 ) (19)

. (16)

0 *L*(*q*, *q*˙, *t*)*dt* (17)

(18)

*WHus*

$$0 = \frac{\partial \mathcal{S}(q', q')}{\partial q'} = p'' - p' \tag{23}$$

which dictates the periodic orbits. We obtain finally the periodic orbit theory formula for the DOS

$$\rho(E) \simeq \overline{\rho(E)} + \operatorname{Im} \sum\_{r \in \operatorname{PO}} \frac{T\_r}{\pi \hbar} \sum\_{n \neq 0} \frac{\exp\left\{ i n \left[ \frac{\mathbf{S}\_r}{\hbar} - \frac{\pi}{2} l\_r \right] \right\}}{\left[ \det((M\_r)^n - 1) \right]^{\frac{1}{2}}} \tag{24}$$

Here the first sum runs over all primitive POs and the *n* sum counts the repetitions of each peridic orbit; *Tr*, *Sr*, and *lr* denote the period, action, and Maslov index of the primary PO, and the matrix *M* stands for the monodromy matrix of the primary PO.

In AKP *m* = 0 sector, the motion is restricted in a fixed plane which includes the heavy axis, and the problem essentially reduces to two dimensional one. (Later on the three dimensional feature is recovered only by the proper choice of the Maslov index [4]). As for AKP unstable periodic orbits, *M* has two eigenvalues *eu* and *e*−*u* (hyperbolic case) and the determinant in (24) is given by

$$\frac{1}{2}\left[\det(M^n - 1)\right]^{\frac{1}{2}} = -i\sinh\left(nu/2\right) \tag{25}$$

#### *2.3.2. Naming of a periodic orbit*

In AKP every PO can be coded by the sign of the heavy axis coordinate when the heavy axis is crossed by it. Note that number of the crossings must be even (2*nc*) for the orbit to close.

In this note we shall denote the PO according to Gutzwiller's identification number along with the Bernoulli sequence of POS. (See Table 1 in ref. [3] which gives a complete list1 of POs up to *nc* = 5 for the anisotropy *γ* = 0.2. ) For instance, PO36(+ + − + + −) is the identification number 6 among *nc* = 3 POs.

#### *2.3.3. The contribution of a periodic orbit to the density of state*

The contribution of a single periodic orbit *r* to the DOS is estimated by a resummation of the sum over the repetition j (after the approximation sinh *x* ≈ *ex*/2),

$$\left. \rho(E) \right|\_{r} \approx T\_{r} \sum\_{m} \frac{u\_{r} \hbar/2}{\left( \mathcal{S}\_{r} - 2 \pi \hbar (m + 1/4) \right)^{2} + \left( u\_{r} \hbar/2 \right)^{2}}. \tag{26}$$

This gives Lorentzian peak at

$$S\_{\Gamma} = 2\pi\hbar(m + l/4) \tag{27}$$

10.5772/55208

31

http://dx.doi.org/10.5772/55208

Classical and Quantum Correspondence in Anisotropic Kepler Problem

**2.4. Scars as observed in the probability distributions and Husimi functions**

periodic orbit FPO (+ −) which reduces the Kepler ellipse orbit in the limit *γ* = 1.

to use the ability of POT predictions (24) to locate the scaring levels.

Let us start exploring the scars in AKP first by investigating the case of the fundamental

In Figure 1, we show the wave function squared in the *µν* plane and the Husimi distribution in the *µ pµ* plane. The FPO is shown by red line and compared with the probability distributions in the in the *µν* plane. At high anisotropy the orbit is largely distorted. Still at chosen energy levels (upper row) we find clear localization around the FPO for both anisotropies. In the lower row we have displayed other energy eigenstates. For these energy levels we see also characteristic probability distribution patterns but not around FPO. Now let us look at the Husimi distributions. In case of energies in the upper diagrams we see very clearly that around the Poincaré section of the FPO (the fixed points) the Husimi functions show clear scars, while in the lower we see anti-scars, the Husimi density is very low at the fixed points. It is clear that Husimi functions are superior observables. In this demonstration of scars we have scanned thousands of energy eigenstates and picked examples. Next task is

**Figure 1.** Scar and anti-scar phenomena with respect to the fundamental periodic orbit. The left set is for the anisotropy *γ* = 0.2 and right for *γ* = 0.6. In each set, the upper and lower row display prominent scar and anti-scar respectively, while the left and right columns exhibit the probability distribution on the *µν* plane and Husimi distribution on the *µ pµ* plane (*ν* = 0). The fundamental orbit is drawn by a red line on the *µν* plane and its Poincaré section on the *µ pµ* plane by red points. The respective eigenvalues are *E*786, *E*787, *E*438, *E*459 in the m = 0, ℓ =even sector. Classical kinematical boundaries are shown by

yellow circles.

*2.4.1. The manifestation of the fundamental FPO*(+ −)

similar to the Bohr-Sommerfeld formula. In AKP the action *S* is given as

$$S\_{\mathcal{I}}(E) = \frac{T\_r}{\sqrt{-2E}}.\tag{28}$$

Hence the peak position of the Lorentzian form in the energy is given by

$$E\_{r,m} = -\frac{1}{2} \left( \frac{T\_r}{2\pi\hbar(m+1)} \right)^2,\tag{29}$$

where Maslov index *l* = 4 for three dimensions is taken.

We are aware that it is meant by (24) that the exact DOS with sharp delta function peaks on the energy axis corresponds to the sum of all PO contributions [20] (assuming convergence).

It is the collective addition of all POs that gives the dos. But, still, it is amusing to observe that the localization of wave functions occurs around the classical periodic orbits as we will see below.

1 In [3, 5] an amazing approximation formula that gives a good estimate of the action of each periodic orbit from its symbolic code is presented. The trace formula has a difficulty coming from the proliferation of POs of long length. This approximation gives a nice way of estimating the sum. The table is created to fix the two parameters involved in the approximation. We thank Professor Gutzwiller for informing us of this formula a few years ago.

### **2.4. Scars as observed in the probability distributions and Husimi functions**

#### *2.4.1. The manifestation of the fundamental FPO*(+ −)

8 Advances in Quantum Mechanics

*2.3.2. Naming of a periodic orbit*

This gives Lorentzian peak at

see below.

identification number 6 among *nc* = 3 POs.

*2.3.3. The contribution of a periodic orbit to the density of state*

*ρ*(*E*)|*r* ≈ *Tr* ∑

sum over the repetition j (after the approximation sinh *x* ≈ *ex*/2),

*m*

similar to the Bohr-Sommerfeld formula. In AKP the action *S* is given as

Hence the peak position of the Lorentzian form in the energy is given by

*Er*,*m* = −1

where Maslov index *l* = 4 for three dimensions is taken.

2

In AKP every PO can be coded by the sign of the heavy axis coordinate when the heavy axis is crossed by it. Note that number of the crossings must be even (2*nc*) for the orbit to close. In this note we shall denote the PO according to Gutzwiller's identification number along with the Bernoulli sequence of POS. (See Table 1 in ref. [3] which gives a complete list1 of POs up to *nc* = 5 for the anisotropy *γ* = 0.2. ) For instance, PO36(+ + − + + −) is the

The contribution of a single periodic orbit *r* to the DOS is estimated by a resummation of the

(*Sr* − 2*πh*¯(*m* + *l*/4))

*Sr*(*E*) = *Tr* √−2*E*

 *Tr* 2*πh*¯(*m* + 1)

We are aware that it is meant by (24) that the exact DOS with sharp delta function peaks on the energy axis corresponds to the sum of all PO contributions [20] (assuming convergence). It is the collective addition of all POs that gives the dos. But, still, it is amusing to observe that the localization of wave functions occurs around the classical periodic orbits as we will

1 In [3, 5] an amazing approximation formula that gives a good estimate of the action of each periodic orbit from its symbolic code is presented. The trace formula has a difficulty coming from the proliferation of POs of long length. This approximation gives a nice way of estimating the sum. The table is created to fix the two parameters involved

in the approximation. We thank Professor Gutzwiller for informing us of this formula a few years ago.

2

*urh*¯ /2

2 + (*urh*¯ /2)

*Sr* = 2*πh*¯(*m* + *l*/4) (27)

. (28)

, (29)

2 . (26)

Let us start exploring the scars in AKP first by investigating the case of the fundamental periodic orbit FPO (+ −) which reduces the Kepler ellipse orbit in the limit *γ* = 1.

In Figure 1, we show the wave function squared in the *µν* plane and the Husimi distribution in the *µ pµ* plane. The FPO is shown by red line and compared with the probability distributions in the in the *µν* plane. At high anisotropy the orbit is largely distorted. Still at chosen energy levels (upper row) we find clear localization around the FPO for both anisotropies. In the lower row we have displayed other energy eigenstates. For these energy levels we see also characteristic probability distribution patterns but not around FPO. Now let us look at the Husimi distributions. In case of energies in the upper diagrams we see very clearly that around the Poincaré section of the FPO (the fixed points) the Husimi functions show clear scars, while in the lower we see anti-scars, the Husimi density is very low at the fixed points. It is clear that Husimi functions are superior observables. In this demonstration of scars we have scanned thousands of energy eigenstates and picked examples. Next task is to use the ability of POT predictions (24) to locate the scaring levels.

**Figure 1.** Scar and anti-scar phenomena with respect to the fundamental periodic orbit. The left set is for the anisotropy *γ* = 0.2 and right for *γ* = 0.6. In each set, the upper and lower row display prominent scar and anti-scar respectively, while the left and right columns exhibit the probability distribution on the *µν* plane and Husimi distribution on the *µ pµ* plane (*ν* = 0). The fundamental orbit is drawn by a red line on the *µν* plane and its Poincaré section on the *µ pµ* plane by red points. The respective eigenvalues are *E*786, *E*787, *E*438, *E*459 in the m = 0, ℓ =even sector. Classical kinematical boundaries are shown by yellow circles.

#### *2.4.2. PO prediction and AKP Scars*