5
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ You should probably look at the branch of math called semiclassical analysis. In particular, I think that the Weyl law may be useful in this context... $\endgroup$ – yuggib Aug 25 '15 at 8:07
  • $\begingroup$ Wow, this is exactly what I've been looking for! The math is over my head at this point, but I'll definitely come back to it in a year or two. $\endgroup$ – knzhou Aug 25 '15 at 23:32
  • 2
    $\begingroup$ This is an interesting question for sure, but reaaaaly broad. $\endgroup$ – DanielSank Sep 5 '15 at 3:12
3
+100
$\begingroup$

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.

--

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.