Don't be intimidated, semiclassical quantization is very simple, and it can be straightforwardly understood from a few examples which lead to the general case.

Consider a particle in a box. The classical motions are reflections off the wall. These make a box in phase space, as the particle goes left, hits the wall, goes right, and hits the other wall. If the particle has momentum p and the length of the box is L, the area enclosed by this motion in phase space is

$$ p L $$

and the condition is that this is an integer multiple of $h=2\pi\hbar$. This gives the momentum quantization condition from quantum mechanics.

For a 1 dimesional system, the rule is that

$$ \int p dx = n h$$

With a possible offset, so that the right-hand side might be $(n+1/2)h$, or $(n+3/4)h$, as appropriate, but the spacing between levels is given by this rule to leading order in h. This rule can be understood from deBroglie's relation--- the momentum at any x is the wavenumber, or the rate of change of the phase of the wavefunction. The condition (in natural units where $h=2\pi$ ) is saying that the phase change as you follow a classical orbit should be an integer multiple of $2\pi$, i.e. that the wave should form a standing wave.

This formula is not exact, because the quantum wave doesn't follow the classical trajectory, but the WKB approximation just takes this as a starting point, and makes a wave whose phase is given by the value of this integral, and whose amplitude is the reciprocal of the square root of the classical velocity.

The reason this works was known already before quantum theory was fully formulated. But to understand it requires familiarity with action-angle variables

### Action-angle variables

Consider an orbit of a particle in one-dimension, with position x and momentum p. You call the area in phase space enclosed by the orbit J, and this is the action. J is only a function of H and it is constant in time (by definition).

The conjugate variable to J is a variable which distinguishes the points of the orbit, and this is called $\theta$. Now you notice that the area in phase space is invariant under canonical transformations (for infinitesimal canonical transformations this is Liouville's theorem), so that the area between the orbits at J and J+dJ is the same as the area in x-p coordinates between J and J+dJ, which is just dJ because that's the definition of J. But this area in J,$\theta$ coordinates is dJ times the period of $\theta$, so $\theta$ has the same period for all J, which I will take to be $2\pi$.

The rate at which $\theta$ increases with time is given by Hamilton's equations

$$ \dot{\theta} = {\partial H\over \partial J} = H'(J) $$

And this is constant over the entire orbit, because H is constant, and so is J. So you learn that $\theta$ increase monotonically at a constant rate at each J, and the time period of $\theta$ is:

$$ T = {2\pi\over H'(J)} $$

### Semiclassical quantization

Suppose you weakly couple this one-dimensional system to electromagnetism. The classical orbital frequency is going to be the frequency of the emitted photons (and double this frequency, and three times this frequency), so that if you want to have discrete photon-emission transitions, you must ensure that emitting a photon of frequency $f={1\over T}$, and taking away energy $hf$ leaves you with a quantum state to fall to. So if there is a quantum state corresponding to a classical motion with one value of J, at energy H(J), there must be another quantum state with energy

$$ H(J) - {2\pi h\over T} = H(J) - H'(J)h \approx H(J-h) $$

in other words, the quantum states must be spaced evenly in J. To this order, this means that there are states at J-h,J-2h,J-3h and so on, and transitions to these states have to reproduce the classical radiation harmonics produced when you weakly couple the thing to electromagnetism.

So the quantization rule is $J=nh$, up to a possible offset. The derivation makes it clear that it is only true to leading order in h. This was Bohr's correspondence argument for the quantization condition.

When you have more than one degree of freedom, and the system is integrable, you have action variables $J_1,J_2...J_n$ and conjugate angle variables periodic with period $2\pi$ each. You can couple any of the degrees of freedom to electromagnetism weakly, and each classical period of the $\theta$ variable in time is

$$T_k = {\partial H \over \partial J_k}$$

so the statement is that for each orbit, each J variable is quantized according to the Bohr rule.

$$ J_k = nh $$

The $J_k$ variable is the area enclosed in the one dimensional projection of the motion in those coordinates where the motion separates into multiperiodic motion (this is the torus of Bar Moshe's answer). This is Sommerfeld's extension of Bohr quantization.

So the integral $\int p dq$ is taken with p and q any conjugate variables which make a period motion. In 1d, there is nothing to do, in multiple dimensions, you just choose variables which separately execute a 1d motion, and in general, you have to find J variables. This procedure doesn't work for classically chaotic systems.

This post imported from StackExchange Physics at 2017-03-13 12:20 (UTC), posted by SE-user Ron Maimon