Planck's constant and phase space in quantum mechanics

Question

During my undergrad physics classes, I've come across several seemingly related phenomena dealing with $h$ and phase space in quantum mechanics.

1. Let $T_x$ be a translation operator by $x$ in position space, and let $T_p$ translate by $p$ in momentum space. Then $T_x$ and $T_p$ commute if $xp$ is a multiple of $h$.
2. Suppose a system of particles can occupy some volume of $2N$-dimensional phase space. Then the density of quantum states (in certain nice cases) is $1/h^{2N}$.
3. The quantization condition of old quantum theory: the legal phase space paths satisfy $\int pdq = nh$.
4. The WKB approximation with two hard walls: stationary states must satisfy $\int pdx = nh/2$.

As indicated by the comment below, these things are all linked by the field of semiclassical analysis. I'd like a brief overview of what semiclassical analysis is and how these connections work.

Answer 1

I) The common starting point is the CCR

$$\tag{1} [\hat{Q},\hat{P}]~=~i\hbar~{\bf 1}.$$

For a general irreducible representation of the CCR (1), see the Stone-von Neumann theorem. The standard Schrödinger position representation reads

$$ \tag{2} \hat{Q}~=~q, \qquad \hat{P}~=~-i\hbar\frac{\partial}{\partial q}. $$

There is a similar Schrödinger momentum representation. The CCR (1) also dictates the overlap between the position and momentum basis

$$\tag{3} \langle p,t \mid q,t \rangle~=~\frac{1}{\sqrt{2\pi\hbar}}\exp\left(\frac{pq}{i\hbar}\right) $$

up to phase factor conventions, cf. e.g. this. Phys.SE post. It follows that the exponentiated operators $$\tag{4} T_a~:=~\exp\left(\frac{ia}{\hbar}\hat{P}\right)\quad \text{and}\quad \tilde{T}_b~:=~\exp\left(\frac{b}{i\hbar}\hat{P}\right)$$ become the translation operators $$\tag{5} T_a\psi(q)~=~ \psi(q+a), \qquad \tilde{T}_b\tilde{\psi}(p)~=~ \tilde{\psi}(p+b). $$ From the CCR (1) and the BCH-like formula $$ \tag{6} e^{\hat{A}}e^{\hat{B}}~=~e^{\hat{C}}e^{\hat{B}}e^{\hat{A}},\qquad \hat{C}~:=~[\hat{A},\hat{B}], $$ which holds if $$ \tag{7} [\hat{A},\hat{C}]~=~0\quad \text{and}\quad [\hat{B},\hat{C}]~=~0, $$ it is straightforward to see that $$\tag{8} \left[ T_a, \tilde{T}_b\right]~=~0 \qquad\Leftrightarrow\qquad ab~\in~ h\mathbb{Z},$$ which is OP's first statement.

II) The TISE in the Schrödinger position representation reads $$ \tag{9} (\hat{P}^2-p(q)^2 )\psi(q)~=~0, \qquad p(q)~:=~ \sqrt{2m(E-V(q))}. $$ The semiclassical WKB expansion $$\tag{10} \psi(q)~=~A(q)\exp\left(\frac{i}{\hbar}S(q)\right) $$ leads to $$\tag{11} \frac{dS(q)}{dq}~=~\pm p(q). $$ The WKB/Bohr-Sommerfeld quantization condition$^1$ $$\tag{12} \oint p(q)~dq ~\in~ h\mathbb{Z} $$ then follows essentially from the fact that the wave function (10) should be single-valued. For a more detailed derivation, see e.g. references given in this Phys.SE post. The WKB/Bohr-Sommerfeld quantization condition (12) shows that in 1D there is roughly one bound state per classically available phase space area divided by Planck's constant $h$. This generalizes to higher dimensions, see e.g. Weyl's law, cf. above comment by user yuggib.

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$^1$ In eq. (12) we have for simplicity ignored the metaplectic correction/Maslov index.