# A New Look at the Quantum Measurement Problem

Z.E. Musielak  Department of Physics, University of Texas at Arlington,
Arlington, TX 76019, USA

###### Abstract

A novel solution to the quantum measurement problem is presented by using a
new asymmetric equation that is complementary to the Schrödinger equation.
Solved for the hydrogen atom, the new equation describes the temporal and
spatial evolution of the wavefunction, and the latter is used to calculate the
radial probability density for different measurements. The obtained results
are consistent with Born’s position measurement postulates. Specific
experimental verification is also proposed.

##  1 Introduction

The measurement problem in nonrelativistic quantum mechanics (NRQM) has been
an active area of research for almost 100 years. Numerous attempts to solve it
(e.g., [1-22]) have been unsuccessful. The proposed ideas are aligned to
different interpretations of NRQM, and they range from the objective
wavefunction collapse theories, such as the Copenhagen and Ghirardi-Rimini-
Weber (GRW) interpretations [2,7-9,13], to the deterministic DeBroglie-Bohm
interpretation with its non-local hidden variables [9,13], or to quantum
decoherence [15-17], and Everett’s many-words interpretation [3,13]. Penrose
[21] proposed a collapse theory that attributes the wavefunction collapse to
the influence of gravity. The role of quantum gravity in solving the
measurement problem was further explored in [22]; however, this requires to
formulate quantum gravity as a fundamental theory of modern physics, which has
not yet been accomplished.

According to the Copenhagen interpretation, quantum measurement involves the
projection of a previously undetermined quantum state into some measurement
eigenstate (e.g., [2,9,13]). This cannot be achieved by the Schrödinger
equation (SE) alone. The reason is that the time-evolution of a quantum state
$|\Psi>=\hat{U}|\Psi_{o}>$, where $\hat{U}$ is the time-evolution operator for
the equation, is continuous and deterministic; however, the projection of
$|\Psi>$ into a measurement eigenstate is discontinuous and indeterministic.
Thus, von Neumann [2] introduced the reduction postulate, which combines the
SE and the collapse of the wavefunction. This leads to ambiguity when a
measurement must be switched on and considered as an interaction [9,13]. A
possible solution to this problem and its underlying ambiguity is proposed in
this paper.

The proposed solution is based on a new asymmetric equation (NAE) [23], whose
existence is guaranteed by the irreducible representations (irreps) of the
Galilean group [24-32]. The NAE is complementary to the SE [33,34] that also
originates from the same irreps [31,32]. The main advantage of the NAE, as
compared to the SE, is that it allows measurements to be directly accounted
for. The effects of the measurements on the evolution of the wavefunction are
investigated by solving the new equation for the hydrogen atom. The obtained
solutions represent the temporal and spatial evolution of the wavefunction,
and they are used to compute the radial probability densities of finding the
electron after the measurement process is completed.

To account for the original electron’s probability density resulting from the
SE, the products of this probablity and those resulting from measurements are
calculated using the NAE. The obtained products are consistent with Born’s
position measurement postulates (e.g., [33,34]). Experimental verification of
the calculated probability densities is proposed and discussed. The presented
results demonstrate that the full description of the wavefunction in NRQM may
require both the Schrödinger equation and its complementary new aymmetric
equation.

The paper is organized as follows: the basic equations resulting from the
Galilean group are given and discussed in Section 2; the governing equation is
presented in Section 3; the temporal and spatial solutions to the governing
equation are given in Sections 4 and 5, respectively; experimental
verification is discussed in Section 6; and Conclusions are given in Section
7.

##  2 Basic equations from the Galilean group

The irreducible representations (irreps) of the Galilean group, $\mathcal{G}$,
are well-known and classified [24-30]. A scalar function $\phi(t,\mathbf{x})$
transforms as one of the irreps of $\mathcal{G}$ if the following eigenvalue
equations $i\partial_{t}\phi=\omega\phi$ and $-i\nabla\phi=\mathbf{k}\phi$ are
satisfied, where $\partial_{t}=\partial/\partial t$, and $\omega$ and
$\mathbf{k}$ are the eigenvalues that label the irreps [31,32]. Using the
eigenvalue equations only two second-order partial differential equations that
are asymmetric in space and time derivatives can be derived. The coefficients
in these equations are expressed as ratios of the labels of the irreps. If
these labels are identified as wave frequencies and wavenumbers, the resulting
asymmetric equations describe propagation of classical waves [35].

However, if the coefficients are determined with the de Broglie relationship
(e.g., [33,34]), then, one of these asymmetric equations becomes the
Schrödinger equation

$\left[i{{\partial}\over{\partial
t}}+\frac{\hbar}{2m}\nabla^{2}\right]\phi_{S}(t,\mathbf{x})=0\ ,$ (1)

and the other can be written as

$\left[\frac{i}{\omega}{{\partial^{2}}\over{\partial
t^{2}}}+\frac{\hbar}{2m}{\mathbf{k}}\cdot\nabla\right]\phi_{A}(t,\mathbf{x})=0\
,$ (2)

which is a new asymmetric equation [23], with $\phi_{S}(t,\mathbf{x})$ and
$\phi_{A}(t,\mathbf{x})$ being the wavefunctions satisfying these equations.
The equations can also be derived from the relativistic energy-momentum
relation by taking the nonrelativistic limit, which shows tha the expression
for the nonrelativistic kinetic energy $E_{k}=p^{2}/2m$ underlies both
equations. Based on the mathematical structure and physical meaning of the SE
and NAE, the equations are complementary to each other, and they describe
different aspects of the evolution of the wavefunction in NRQM.

There are several important differences between the SE and NAE. The most
prominent one is the difference in time derivatives. By being first-order in
time, the time solutions to the SE are also the solutions to the continuity
equation for probability, which is also first-order in time. This means that
the SE is invariant with respect to unitary transformations, and that NRQM
based on the SE is a unitary theory. On the other hand, the NAE being second-
order in time may have solutions that do not satisfy the continuity equation
and, as a result, the NAE may be used to describe some non-unitary processes
in NRQM. Thus, the SE (unitary) $\rightarrow$ the NAE (non-unitary), with the
latter describing processes as the measurement problem or the quantum jumps
representing absorption or emission of radiation by atoms (e.g., [33,34]);
only the former is considered in this paper.

Another difference between the SE and NAE is the presence of the eigenvalues
$\omega$ and $\mathbf{k}$ in the latter. To describe any physical event or
process by the NAE requires specifying both eigenvalues. By identifying
$\omega$ and $\mathbf{k}$ as the characteristic frequency and wavevector of
dark matter, some attempts were made to formulate a quantum theory of cold
dark matter [36] and galactic cold dark matter halos [37]; however, the
formulated theories require observational verification of their predictions.
In this paper, the eigenvalues are identified with measurements performed on a
quantum system, which is a hydrogen atom in its basic 1s state. The changes in
the temporal behavior of the wavefunction, and in the radial probability
densities of the electron resulting from the performed measurements are
presented and discussed.

##  3 Governing equation and measurements

The labels of the irreps, $\omega$ and $\mathbf{k}$, which appear explicitly
in the NAE can be used to specify interactions between a quantum system and an
apparatus that performs measurements on this system. Denoting
$\omega=\omega_{ap}$, $E_{ap}=\hbar\omega_{ap}$, and
${\mathbf{k}}={\mathbf{k_{ap}}}$, with
${\mathbf{k_{ap}}}={\hat{k}(1/\lambda_{ap}})$, where the parameters
$\omega_{ap}$ and $\lambda_{ap}$ may represent frequency and wavelength of
electromagnetic (EM) waves used in the measurements, respectively, or be
determined by temporal and spatial scales associated with the measuring
process. Then, the eigenvalue equations for the energy,
$\hat{E}=i\hbar\partial_{t}$, and momentum, $\mathbf{\hat{P}}=-i\hbar\nabla$,
operators acting on the wavefunction $\phi_{A}$ are given by
$\hat{E}\phi_{A}=\sqrt{E_{ap}E}\ \phi_{A}$ and
$(\hbar\mathbf{k_{ap}}\cdot\mathbf{\hat{P}})\phi_{A}=(\mathbf{p_{ap}}\cdot\mathbf{\hat{P}})\phi_{A}=p_{ap}p\
\phi_{A}$, with $E=\hbar\omega$, $p=\hbar k$, and $p_{ap}=\hbar k_{ap}$ [37].

The new asymmetric equation in the spherical variables, with the
$r-$dependence only, the potential $V(r)$, and the effects of measurements
accounted for by the presence of $\omega_{ap}$ and ${\mathbf{k_{ap}}}$,
becomes

$\left[\frac{i\hbar}{\omega_{ap}}\left({{\partial^{2}}\over{\partial
t^{2}}}\right)+\frac{\hbar^{2}}{2m}({\mathbf{k}_{ap}}\cdot\nabla)+V(r)\right]\phi_{A}(t,\mathbf{r})=0\
,$ (3)

which is complementary to the SE with the potential to be specified [33,34].

The Schrödinger equation with the Coulomb potential,
$V(r)=-e^{2}/(4\pi\epsilon_{o}r)$, describes a hydrogen atom by giving the
solutions for the wavefunction in terms of Laguerre’s polynomials (the radial-
dependence) and spherical harmonics (the angular-dependence), and predicts
correctly its quantum energy levels given by
$E_{n}=-(m/2n^{2}\hbar^{2})(e^{2}/4\pi\epsilon_{o})^{2}$. The results given by
the SE descibe the electron’s unitary dynamics and they are valid prior to
making any measurement.

As pointed out in Introduction, all previous attempts to account for
measurements by using the SE have failed, including von Neumann’s approach
with the reduction postulate because of its ambiguity. To remove this
ambiguity, it is suggested herein that at the moment when a measuring
apparatus is applied to a quantum system, the system is described by the NAE.
To verify this suggestion, Eq. (3) is now solved for a hydrogen atom in its
basic 1s state, and the temporal and spatial behavior of the wavefunction
resulting from the measurement process is calculated. Then, the spatial
solutions are used to compute changes in the radial probabilty density caused
by measurements.

However, the probability densities resulting from the measurements, and
calculated with the NAE, do not account for the previous electron’s unitary
dynamics. To take this into consideration, the probabilities resulting from
the measurements are multipled by the original probablility density obtained
for the electron from the SE. The resulting probabliltes demonstrate that the
original state of the electron is projected into the observable eigenstate as
suggested by Born’s measurement postulates (e.g., [33.34]).

##  4 Time-dependent solutions and their physical meaning

To solve Eq. (3), the spherical variables are separated into the temporal and
spatial (radial only) components, $\phi_{A}(t,\mathbf{r})=\chi(t)\
\eta(\mathbf{r})$, and $-\mu^{2}$ is chosen as the separation constant. To
relate this approach directly to the hydrogen atom problem and its description
by the SE, it is assumed that $V(r)$ is the Coulomb potential and that
$-\mu^{2}=E_{n}$, which gives

${{d^{2}\chi}\over{dt^{2}}}+i\left(\frac{E_{ap}}{2m}\right)\left(\frac{1}{na_{o}}\right)^{2}\chi=0\
,$ (4)

and

${{d\eta}\over{dr}}+\frac{1}{(\mathbf{\hat{k}_{ap}}\cdot\mathbf{\hat{r}})k_{ap}\
a_{o}}\left(\frac{1}{n^{2}a_{o}}-\frac{2}{r}\right)\eta=0\ ,$ (5)

where $E_{ap}=\hbar\omega_{ap}$ and the Bohr radius
$a_{o}=(4\pi\epsilon_{o}\hbar^{2})/me^{2}=\hbar/(mc\alpha)$, with $\alpha$
being the fine structure constant. It must be noted that both $\omega_{ap}$
and $k_{ap}$ are specified by interactions between a measuring apparatus and a
quantum system.

Using $i=(1/\sqrt{2}+i/\sqrt{2})^{2}$, the solutions to Eq. (4) are

$\chi(t)=C_{\pm}\exp{\left[\pm\
i\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)\sqrt{\frac{E_{ap}}{2m}}\frac{t}{na_{o}}\right]}\
,$ (6)

where $C_{\pm}$ are the integration constants corresponding to the $\pm$
solutions. The solutions require that a measuring apparatus interacts with a
quantum system, that is $\omega_{ap}\neq 0$ as otherwise $\chi=C_{\pm}$, and
the temporal evolution of the wavefunction must be described by the SE. Both
solutions with $C_{+}$ and $C_{-}$ are physical and they correspond to
$t\rightarrow+\infty$ and $t\rightarrow-\infty$, respectively, which shows
that the solutions are symmetric with respect to $t=0$. In the following, only
the solution with $C_{+}$ is considered because $t=0$ is the intital time at
which measurement begins, and $t>0$ when it is performed.

The real part of the solution is

${\cal{R}\it
e}[\chi(t)]=C_{+}\cos{\left(\sqrt{\frac{E_{ap}}{m}}\frac{t}{2na_{o}}\right)}\exp{\left(-\sqrt{\frac{E_{ap}}{m}}\frac{t}{2na_{o}}\right)}\
.$ (7)

The arguments of the cosine and exponential functions can be estimated when
$\omega_{ap}$ is given by specifying a measuring apparatus. Consider a quantum
microscope designed to measure the orbital structure of the hydrogen atom
(e.g., [38]). Then, $\omega_{ap}$ is the frequency of electromagnetic (EM)
waves used to perform the measurement. Let this frequency be either the
optical, ultraviolet, or X-ray part of the EM spectrum, then the factor
$(\sqrt{E_{ap}/m}/a_{o})$ in the arguments of the cosine and exponential
functions becomes either $\sim 10^{15}$ $s^{-1}$, $\sim 10^{16}$ $s^{-1}$, or
$\sim 10^{17}$ $s^{-1}$, respectively. As a result, the arguments are very
large and the exponential function decays very rapidly for any $t>0$, while
the cosine function remains periodic despite its large argument. The time of
this decay of the wavefunction decreases with increasing frequency of the used
EM waves.

Since the argument of the exponential function is very large, the temporal
part of the wavefunction reaches zero on a very short time scale that ranges
from $\sim 10^{-17}$ $s$ to $\sim 10^{-15}$ $s$. The rapid decay of the
wavefunction in time predicted by the NAE and its temporal solutions may be
identified as the duration of the measurement after which the electron resumes
its original orbital. If the decay is considered ’instantaneous’, then
${\cal{R}\it e}[\chi(t)]$ may represent the wavefunction that impinges on a
screen or photographic film in experiments with double-slits, or diffraction
of electrons through a narrow aperture, and be identified with the
’instantaneous collapse’ of the wavefunction (e.g., [39]).

In general, it is expected that the time required for the wavefunction
collapse is shorter when $m$ increases (e.g., [9,13,33]). The arguments of the
periodic and exponential parts of the solution ${\cal{R}\it e}[\chi(t)]$ are
the same, and they depend explictly on the mass of electron $m$. Thus, with
$a_{o}$ being a function of $m$, the arguments depend on $\sqrt{m}$, whose
larger value causes the exponential function to decrease faster, and shortens
the time of the wavefunction collapse.

##  5 Time-indepedent solutions and probability densities

Having obtained the solutions for the temporal evolution of the wavefunction,
the solutions for its spatial behavior are now derived, and the resulting
probabilty densities are calculated. Defining
$\beta_{ap}=[(\mathbf{\hat{k}_{ap}}\cdot\mathbf{\hat{r}})k_{ap}\ a_{o}]^{-1}$,
the solution to Eq. (5) can be written as

$\eta(r)=\eta_{0}\ r^{2\beta_{ap}}\
\exp{\left[-\left(\frac{\beta_{ap}}{n^{2}}\right)\frac{r}{a_{o}}\right]}\ ,$
(8)

where the integration constant $\eta_{0}$ normalizes the wavefunction and is
evaluated from

$4\pi\int_{0}^{\infty}r^{2}|\eta(r)|^{2}dr=1\ ,$ (9)

showing that $\eta_{0}$ depends on $\beta_{ap}$ and on the principal quantum
numbers $n$; thus, $\eta_{o}=\eta_{o,n}(\beta_{ap})$. Moreover, in spherical
symmetry $\mathbf{\hat{r}}$ can always be aligned with
$\mathbf{\hat{k}_{ap}}$, which means that
$\mathbf{\hat{k}_{ap}}\cdot\mathbf{\hat{r}}=1$ and $\beta_{ap}=(k_{ap}\
a_{o})^{-1}=\lambda_{ap}/a_{o}$, where $\lambda_{ap}$ is the wavelength of EM
waves used in the measuring process.

The radial probability density in a spherical shell volume element is given by

${\cal{P}}_{n}(r)=r_{max}\frac{dP(r)}{dr}=\eta^{2}_{o}(\beta_{ap})\left(\frac{r}{r_{max}}e^{-r/r_{max}}\right)^{4\beta_{ap}+2}$
(10)

where $\eta^{2}_{o}(\beta_{ap})=1/I(\beta_{ap})$. Defining $x=r/r_{max}$, the
integral $I(\beta_{ap})$ becomes

$I(\beta_{ap})=\int_{0}^{\infty}(x\ e^{-x})^{4\beta_{ap}+2}\ dx\ ,$ (11)

and it can be evaluated after $\beta_{ap}$ is specified. Moreover, $r_{max}$
is the most probable radius defined as

$r_{max}=n^{2}a_{o}\left(2+\frac{1}{\beta_{ap}}\right)=n^{2}a_{o}\left(2+\frac{a_{o}}{\lambda_{ap}}\right)\
.$ (12)

The effects of measurements are given by $\beta_{ap}\neq 0$ (or
$\lambda_{a}\neq 0$), which is required for the NAE to represent the radial
wavefunction and its evolution. In the absence of measurements, the
wavefunction’s evolution is described by the SE.

To compare the obtained results directly to the radial probability densities
of a hydrogen atom given by the time-independent solutions to the SE, it is
convenient to introduce $r_{a}=r/a_{o}$. Then, the radial probability density
${\cal{P}}_{n}(r_{a})$ given by Eq. (10) can be written as

${\cal{P}}_{n}(r_{a})=\frac{dP(r_{a})}{dr_{a}}=\eta^{2}_{o,n}(\beta_{ap})\
r_{a}^{4\beta_{ap}+2}\ e^{-2(\beta_{ap}/n^{2})r_{a}}\ ,$ (13)

where $\eta^{2}_{o,n}(\beta_{ap})=1/I_{n}(\beta_{ap})$, and the integral
$I_{n}(\beta_{ap})$ can be evaluated by using

$I_{n}(\beta_{ap})=\int_{0}^{\infty}r_{a}^{4\beta_{ap}+2}\
e^{-2(\beta_{ap}/n^{2})r_{a}}=\frac{1}{(2\beta_{ap})^{4\beta_{ap}+3}}\
\Gamma({4\beta_{ap}+3})\ ,$ (14)

which is valid if ${\cal{R}\it e}(2\beta_{ap})>0$ and ${\cal{R}\it
e}(4\beta_{ap}+2)>-1.$

Figure 1: The radial probability density, ${\cal{P}}_{1}(r_{a})$, is plotted
versus the ratio of radius to the Bohr radius, $r/a_{o}$, for $n=1$ and
$\beta_{ap}=\lambda_{ap}/a_{o}$ that ranges from X-rays to the ultraviolet and
visible parts of the EM spectrum. The theoretically predicted
${\cal{P}}_{1}(r_{a})$ is compared to the radial probability density,
${\cal{P}}_{1s}(r_{a})$, for the hydrogen $1s$ orbital.

Using Eqs (13) and (14), the radial probability density ${\cal{P}}_{1}(r_{a})$
is calculated for $n=1$ and for different values of $\beta_{ap}$, which varies
from $50$ to $9000$. This range of $\beta_{ap}$ corresponds to
$\lambda_{ap}=50\ a_{o}$ and $100\ a_{o}$ (X-rays), $\lambda_{ap}=200\ a_{o}$
and $800\ a_{o}$ (ultraviolet), and $\lambda_{ap}=8000\ a_{o}$ and $9000\
a_{o}$ (visible); this range can be extended to the other parts of the EM
spectrum. Since the selected values of $\beta_{ap}$ are positive integers, the
function $\Gamma({4\beta_{ap}+3})=({4\beta_{ap}+2})!$ in Eq. (14). The
obtained results are presented in Fig. 1 showing also the radial probability
density, ${\cal{P}}_{1s}(r_{a})=4r_{a}^{2}\ \rme^{-2r_{a}}$, for the hydrogen
$1s$ orbital plotted for comparison.

The maxima of ${\cal{P}}_{1}(r_{a})$ plotted in Fig. 1 are at $r=r_{max}$,
where the latter depends on $\beta_{ap}$ (see Eq. 12). This dependence on
$\beta_{ap}$ or $\lambda_{ap}$ is reflected in Fig. 1 as small shifts towards
the larger values of $r_{a}$ for shorter wavelengths, such as X-rays and UV.
However, in the limit $\lambda_{ap}\rightarrow\infty$, one finds
$r=r_{max}=2a_{o}$, and

${\cal{P}}_{1}(r_{a})=\lim_{\lambda_{ap}\rightarrow\infty}\eta^{2}_{o,1}(\lambda_{ap})r_{a}^{4\lambda_{ap}/a_{o}+2}\
e^{-2(\lambda_{ap}/a_{o})r_{a}}=\delta(r-2a_{o})\ ,$ (15)

where $\delta(r-2a_{o})$ is the Dirac delta function. This confirms that for
very long wavelengths, ${\cal{P}}_{1}(r=2r_{a})=1$, which is the classical
limit of the measuring process. The main result is that a quantum particle
interacting with a measuring apparatus gets confined into a measurement
eigenstate, or in a well-defined position at $r=2a_{o}$ as originally
suggested by Born’s position measurement principles (e.g., [33,34]).

Figure 2: The product of the radial probability densities,
${\mathfrak{P}}(r_{a})={\cal{P}}_{1}(r_{a})\times{\cal{P}}_{1s}(r_{a})$,
plotted versus the ratio of radius to the Bohr radius, $r/a_{o}$, for $n=1$,
and $\beta_{ap}=\lambda_{ap}/a_{o}$that ranges from X-rays to the visible part
of the EM spectrum.

The results presented in Fig. 1 are obtained by solving the spatial part of
the NAE for specified wavelengths of EM waves used in the measuring process.
However, the results do not take into account the initial state of the
electron on its 1s orbital. The radial probability density for the 1s orbital,
${\cal{P}}_{1s}(r_{a})$, given by the SE is also shown in Fig. 1. In order to
account for the electron’s initial probablity density on the $1s$ orbital, the
product
${\mathfrak{P}}(r_{a})={\cal{P}}_{1s}(r_{a})\times{\cal{P}}_{1}(r_{a})$ is
calculated and plotted in Fig. 2.

All the computed probability densities in Fig. 2 are more narrow and centered
at $r=2\ a_{o}$ as compared with those in Fig. 1. This is an interesting
result as it demonstrates how the original electron’s probability density on
the $1s$ orbital has changed because of the measurement process. Thus, the
results of Fig. 2 are in better agreement with Born’s position measurement
postulates than those given in Fig. 1, which shows that the electron’s initial
probability density calculated by the SE must also be accounted for in the
procedure that describes the measurements.

The above procedure and calculations can also be performed for the principal
quantum numbers $n=2$, $3$, $4$, and so on, which will give different values
of $r_{max}$, and different probability densities; such calculations will be
presented elsewhere.

##  6 Experimental verification

The results presented in Figs 1 and 2 demonstrate that the electron originally
located on the 1s orbital, with its most probable radius at $r=a_{o}$, is
shifted to its most probable radius at $r=2\ a_{o}$ as a result of the
measurements. This means that the electron interacting with a measuring
apparatus gets confined into a measurement eigenstate. Comparison of the
original electron’s probability density ${\cal{P}}_{1s}(r_{a})$ on the orbital
$1s$ (see Fig. 1) to ${\mathfrak{P}}(r_{a})$ plotted in Fig. 2 shows that as a
result of the measurement the electron is confined to a well-defined position
at $r=2a_{o}$, and that its probability density is very narrow and sharply
centered at this position, which is consistent with Born’s position
measurement principles (e.g., [33,34]). The time scale for the existence of
this measurement eigenstate is given by Eq. (7), which represents the duration
of this eigenstate.

The above theoretical results can be verified experimentally by using a
quantum microscope similar to that designed by an international team of
researchers [38], who used it to measure the orbital structure of Stark states
in an excited hydrogen atom (see their Fig. 3). They reported a similar trend
in shifting the maxima and broadening the density distributions for shorter
wavelengths as those presented in Fig. 2. There are several other measuring
methods developed to study the quantum-classical correspondence [39-43], but
some of these methods may not be suitable to observe single orbitals in a
hydrogen atom [42].

##  7 Conclusions

In this paper, a new asymmetric equation, which is complementary to the
Schrödinger equation, is used to investigate the measurement problem of
quantum mechanics. The obtained results demonstrate that while the Schrödinger
equation describes the evolution of the wavefunction prior to any measurement,
the new asymmetric equation represents the behavior of the wavefunction during
the measurement process. The temporal solutions to the new asymmetric equation
give a time-scale for the duration of the measurement eigenstate, whose
probability density is determined from the spatial solutions.

To describe the transition from unitary (reversible) representation of the
electron on its $1s$ orbital, which is given by the Schrödinger equation, to
its non-unitary (irreversible) evolution resulting from the measurement, which
is given by the new asymmetric equation, the product of both probabilty
densities is calculated. The main results are that the time-scales are or the
order of $10^{-17}$ \- $10^{-15}$ s, depending on the frequency of the EM
waves used in the measurement, and that the computed probability densities are
sharply centered around $r=2\ a_{o}$, which is consitent with Born’s position
measurement postulates.

The predicted radial probability densities resulting from the measurement
process can be verified experimentally by using a quantum microscope (e.g.,
[38]). Based on the presented results, it is proposed that while the unitary
evolution of the wavefunction is described by the Schrödinger equation, its
non-unitary behavior caused by measurements is represented by the new
asymmetric equation; this means that both equations are required in
nonrelativistic quantum mechanics to fully describe the behavior of the
wavefunction.

Acknowledgment: The author thanks Dora Musielak for valuable comments on the
earlier version of this manuscript.

References

## References

  * [1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 (1935) 777
  * [2] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Uni. Press, Princeton, 1955
  * [3] H. Everett, Rev. Mod. Phys. 29 (1957) 454
  * [4] E.P. Wigner, Am. J. Phys. 31 (1963) 6
  * [5] H.D. Zeh, Found. Phys. 1 (1970) 69
  * [6] L.E. Ballentine, Rev. Mod. Phys. 42 (1970) 358
  * [7] G-C. Ghirardi, A, Rimini, and T. Weber, Lett. Nuovo Cimento 27 (1980) 293
  * [8] G-C. Ghirardi, A, Rimini, and T. Weber, Phys. Rev. D 34 (1986) 470
  * [9] J.A. Wheeler and W.H. Zurek, Quantum Theory of Measurement, Princeton Uni. Press, Princeton, 1983
  * [10] N.G. Van Kampten, Physica A 153 (1988) 153
  * [11] J.S. Bell, Physics World 3 (1990) 33
  * [12] T. Maudlin, Topoi 14 (1995) 7
  * [13] P. Mittelstaedt, The Interpretation of Quantum Mechanics and the Measurement Process, Cambridge Uni. Press., Cambridge, 1998
  * [14] M. Danos, and T.D. Kieu, Int. J. Mod. Phys. E 8 (1999) 257
  * [15] W.H. Zurek, Rev. Mod. Phys. 75 (2003) 715
  * [16] E. Joos, H.D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I.-O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer-Verlag, Berlin-Heidelberg, 2003
  * [17] M. Scholsshauer, Rev. Mod. Phys. 76 (2004) 1267
  * [18] A.E. Allahverdyan, R. Balian, and T.M. Nieuwenhuizen, Phys. Reports 525 (2013) 1
  * [19] L. Masanes, T.D. Galley, and M.P. Müller, Nature Comm. 10 (2019) 1
  * [20] J.R. Hance, and S. Hossenfelder, arXiv:2206.10445v3 [quant-ph] 1 Oct 2022
  * [21] R. Penrose, Gen. Relat. and Gravit. 28 (1996) 581
  * [22] E. Shaghoulian, arXiv:2305.10635v1 [hep-th] 18 May 2023
  * [23] Z.E. Musielak, Int. J. Mod. Phys. A 36 (2021) 2150042 (12pp)
  * [24] E.P. Wigner, Ann. Math. 40 (1939) 149
  * [25] E. Inönu and E.P. Wigner, Nuovo Cimiento 9 (1952) 705
  * [26] V. Bargmann, Ann. Math. 59 (1954) 1
  * [27] J.-M. Levy-Leblond, Comm. Math. Phys. 6 (1967) 286
  * [28] J.-M. Levy-Leblond, J. Math. Phys. 12 (1969) 64
  * [29] Y.S. Kim and M.E. Noz, Theory and Applications of the Poincaré Group, Reidel, Dordrecht, 1986
  * [30] S. Sternberg, Group Theory and Physics, Cambridge Uni. Press, Cambridge, 1994
  * [31] Z.E. Musielak and J.L. Fry, Ann. Phys. 324 (2009) 296
  * [32] Z.E. Musielak and J.L. Fry, Int. J. Theor. Phys. 48 (2009) 1194
  * [33] E. Merzbacher, Quantum Mechanics, Wiley & Sons, Inc., New York, 1998
  * [34] J.E. House, Fundamentals of Quantum Mechanics, Academic Presss, Cambridge, MA, 2017
  * [35] Z.E. Musielak, Adv. Math. Phys., (2023) Article ID 5736419 (14 pages)
  * [36] Z.E. Musielak, Int. J. Mod. Phys. A 37 (2022) 2250137 (10 pages)
  * [37] Z.E. Musielak, Int. J. Mod. Phys. D 32 (2023) 2350083 (13 pages)
  * [38] A.S. Stodolna, A. Rouzée, F. Lépine, S. Cohen, F. Robicheaux, A. Gijsbertsen, J.H. Jungmann, C. Bordas and M.J.J. Vrakking, Phys. Rev. Let. 110 (2013) 213001
  * [39] J. Baggott, The Meaning of Quantum Theory, Oxford Uni. Press, Oxford, 1992
  * [40] I.E. McCarthy and E. Weigold, Rep. Progr. Phys. 51 (1988) 299
  * [41] L.D. Noordam and R.R. Jones, J. Mod. Optics 44 (1997) 2515
  * [42] J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J.C. Kieffer, P.B. Corkum and D.M. Villeneuve, Nature 432 (2004) 867
  * [43] D. Shafir, Y. Mairesse, D.M. Villeneuve, P.B. Corkum and N. Dudovich, Nature Phys. 5 (2009) 412