The connection between classical phase space and quantum multiplicity

Question

I am aware of the relationship $N = V/h^n$ where $N$ is the quantum multiplicity, $n$ is the number of position (or momenta) degrees of freedom, $V$ is the volume of classical phase space and $h$ is planck's constant. What I don't know, however, is why this formula seems to work. I've seen the derivation (several ways, in fact, which includes answers here on stack and in stat mech books), I've also seen several examples where I could verify this relation (infinite well, harmonic oscillator), but it's still a complete mystery to me why this should be true. I know that quantum mechanics should approach classical mechanics in the high quantum number limit but I just don't see specifically where this limit is built in, if you get what I mean. More precisely i'm looking for something along the following lines: if somebody can point me to an expression which should obviously tend to a classical value in the classical limit. I need a way to "see" how classical expressions emerge from quantum ones. Note that I'm aware of the connection between poisson brackets/commutators as well as Ehrenfest theorems which provide some connection but i'm looking for one which makes the aformentioned expression obvious.

Answer 1

The standard intuitive path to the classical limit, or, conversely, to quantization, often goes through the phase space formulation of quantum mechanics.

The analog of the Liouville probability density—-let's take n=1 for one degree of freedom and easily generalize later—-is the Wigner quasi probability density, normalized $$ \int\!\!dqdp~ f(q,p)=1, $$ but also, most importantly, $$ -2/h\leq f(q,p)\leq 2/h . $$ (Don't worry now about how this was proven--the Cauchy-Schwartz inequality; nor why the lower bound is not zero: f may go negative... it's quantum mechanics...) A further feature of such pure states turns out to be $ \int\!\!dqdp~ f(q,p)^2= 1/h $.

For the seat-of-the-pants intuition invited here, one is cavalier with small numerical factors.

That means that if you have a Dirac-δ Liouville density for one state for 1 degree of freedom, this cannot be quantized just like that to a Wigner function. Classical mechanics may be spiked and precise/certain, but QM may not, as you just saw: it has to be fuzzy and stubby, bounded above and below by 2/h as you saw above.

The best you can do to localize your Wigner function is to squish all the state's probability density to an h/2-tall pillbox of base about V=h in phase space, so the normalization condition will be comported with, $$ \int\!\!dqdp~ f(q,p)=V ~ 2/h=1. $$ (Don't worry about the factor of 2 for the time being.) You can't squish it to a smaller area/volume. By contrast, note how the classical limit $h\to 0$ obviates these restrictions and allows for a δ-function.

A phase-space Gaussian of this type is, in fact, the ground state of the quantum harmonic oscillator, the optical vacuum. You may see all these arguments and oscillator features in this review or video. These are all consequences of the uncertainty principle as it manifests itself in this formulation, in an unexpected characteristic way.

• So, the minimum phase-space area/volume that you may jam a quantum state in is V/h.

By using a fundamental theorem in the formulation (de Bruin, Cartwright), you may also prove that this is also roughly the maximum area exhibiting compactly negative values for the Wigner function f: all negative features hide in such small areas in phase space; and dissolve/turn positive upon (low-pass, Weierstrass transform) filtering them with a Gaussian $\propto \exp (-(p^2+q^2)/2\hbar )$ or wider.