A body undergoing periodic motion in an orbit of quantum number $n$ will have a period $T$, determined by $$T=\oint \frac{ds}{v}=\oint \frac{ds}{\sqrt{\frac{2}{m}(E-V)}}$$ Where $ds$ is an infinitesimal displacement, $v$ is the body's speed, $m$ is its mass, $E$ its total energy, and $V$ its potential energy. Likewise, it'll have an abbreviated action over the whole period equal to $$J=\oint\vec p\cdot d\vec s =\sum_i\oint p_idq_i=\oint\sqrt{2m(E-V)}ds$$ It can be easily seen that $$T=\dfrac{dJ}{dE}$$ Now, according to Bohr, in the limit of large quantum numbers (corresponding to large vibrations) the behaviour of the body should approach its classical behavior. So the frequency of light emitted by a body as it drops from state $n$ to a lower state should be an integer multiple of the frequency at which the body moves in its periodic motion. Since the lowest frequency of light is emitted when the body drops to the state directly below it, that frequency m,ust correspond to the body's frequency of motion in the classical limit. So, according to the Planck Hypothesis: $$f_n\approx\frac{E_n-E_{n-1}}{h}$$ Where $h$ is Planck's constant. Replacing: $$\dfrac{dE}{dJ}\approx\frac{1}{h}\dfrac{dE}{dJ}(J_n-J_{n-1})$$ Cancelling out $\dfrac{dE}{dJ}$ we obtain $$J_n-J_{n-1}=h$$ From this we obtain that action is quantized $$J=\oint\vec p\cdot d\vec s =\sum_i\oint p_idq_i=nh$$ where $n$ is an integer. But the Old Quantum Condition states that $$\oint p_i dq_i=nh$$ Meaning that the action for every generalized coordinate and monetum is quantized, with the $n$ for each coordinate is a quantum number. How does one go from The total action over one period of a body is quantized to The action for each individual coordinate is quantized?

  • $\begingroup$ Separation of variables assuming the system is separable, c.f. Landau Quantum Mechanics Sec. 48. $\endgroup$ – bolbteppa Jan 20 '18 at 2:10
  • $\begingroup$ But how would you go about proving that for any system of canonical coordinates the integral of one of the momenta over its corresponding coordinate over one oscillation will always equal an integer multiple of Planck's Constant? $\endgroup$ – user140323 Jan 20 '18 at 2:14
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    $\begingroup$ "In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates $q_i$ in terms of which the motion is periodic." en.wikipedia.org/wiki/Old_quantum_theory#Basic_principles en.wikipedia.org/wiki/Action-angle_coordinates to justify this method you need to set up the quasi-classical approximation for a system whose Hamiltonian i.e. Schrodinger equation, is separable as is done e.g. in the chapter of the referenced book, as where did anything in your post actually come from? $\endgroup$ – bolbteppa Jan 20 '18 at 2:53
  • $\begingroup$ I figured this derivation out myself, based on the derivation of Bohr's quantized angular momentum I saw on wikipedia and elsewhere here, but I have seen a derivation which was pretty much the same in another StackExchange question. $\endgroup$ – user140323 Jan 20 '18 at 4:00
  1. As user bolbteppa correctly points out in the comments, the old quantum theory only works if we impose some kind of integrability/separability condition. Let us for simplicity assume that the classical autonomous Hamiltonian system with Hamiltonian $H(J)$ has angle-action variables $$(w^1,\ldots,w^n,J_1,\ldots,J_n)\in (\mathbb{S}^1)^n\times \mathbb{R}^n.\tag{1}$$ Hamilton's characteristic function is then $$W(w,J)~=~\sum_{k=1}^nw^kJ_k.\tag{2}$$

  2. A bit oversimplified, a wavefunction of the form $$ \psi~=~\exp\left(\frac{i}{\hbar}W\right)\tag{3} $$ then satisfies the TISE $$H\left(\frac{\hbar}{i}\frac{\partial}{\partial w}\right) \psi~=~H(J) \psi.\tag{4}$$ The requirement of single-valuedness of the wavefunction $\psi$ along the tori $$w^k\sim w^k +1, \qquad k\in\{1,\ldots, n\},\tag{5}$$ then leads to the Bohr-Sommerfeld quantization rule $$ J_k~\in~h\mathbb{Z}, \qquad k\in\{1,\ldots, n\},\tag{6} $$ which OP requested. This is oversimplified in the sense that it does not explain the metaplectic correction/Maslov indices, which modifies eq. (6).

  3. A more rigorous analysis investigates single-valuedness, turning points, & boundary conditions of the wavefunction $\psi$ as a function of positions $(q^1,\ldots q^n)$ [rather than as a function of angles $(w^1,\ldots w^n)$]. For references, see this Phys.SE post and links therein.


Okay, I think I finally got it! Thanks bolbteppa!

A body which oscillates may oscillate with different periods in different degrees of freedom, or coordinates. One such example is a precessing orbit. Now the period of motion in coordinate $i$ is $$T_i=\oint\frac{dq_i}{\dot q_i}$$ Which, according to Hamiltonian Mechanics is $$\oint\frac{dq_i}{\left(\dfrac{\partial H}{\partial p_i}\right)}$$ The action over a full period from just one coordinate $J_i$ is $$J_i=\oint p_i dq_i$$ The derivative of $J_i$ with respect to $H$ is $$\dfrac{\partial J_i}{\partial H}=\oint \dfrac{\partial p_i}{\partial H}dq_i=\oint\frac{dq_i}{\left(\dfrac{\partial H}{\partial p_i}\right)}$$

So $T_i$ is equal to $\dfrac{\partial J_i}{\partial H}$, or the frequency $f_i$ is equal to $\dfrac{\partial H}{\partial J_i}$

From this I can just go through the math from my question: $$f_i(n)=\frac{H(J_n)-H(J_{n-1})}{h}$$ $$\dfrac{\partial H}{\partial J_i}=\frac{1}{h}\dfrac{\partial H}{\partial J_i}(J_i(n)-J_i(n-1))$$ $$J_i(n)-J_i(n-1)=h$$ $$J_i=\oint p_i dq_i=nh$$ Thanks for your help!


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