Classical Limits of Quantum Mechanics

$ℏ \to 0$ .

Classical	Quantum
$\frac{d A}{d t} = [A, H]_{c l a s s i c a l}$	$\frac{d A_{H}}{d t} = \frac{i}{ℏ} [H, A_{H}]$

Where $[A, H]_{c l a s s i c a l}$ are poisson brackets, $\frac{\partial H}{\partial q} \frac{\partial A}{\partial p} - \frac{\partial A}{\partial q} \frac{\partial H}{\partial p}$ .

Dirac’s rule: $[]_{c l a s s i c a l} \Leftrightarrow []_{Q M} / (i ℏ)$ . Some things in QM cannot be expressed classically: i.e. spin.

CM	QM
No Uncertainty	Uncertainty
Zero Energy of Oscillator	Finite Ground State of Oscillator
Continuous Spectra of Oscillator	Discrete Spectra of Oscillator
Hamilton’s Principle Function	QM Wavefunction Phase

$Ψ (\vec{x}, t) = \sqrt{P (\vec{x}, t)} \exp (\frac{i}{ℏ} S (\vec{x}, t))$ , $P (\vec{x}, t) = | Ψ (\vec{x}, t) |^{2}$ .

$\vec{j} (\vec{x}, t) = \frac{ℏ}{m} ℑ (Ψ^{*} \vec{\nabla} Ψ) = \frac{ℏ}{m} ℑ (\sqrt{P} \exp (- i S / ℏ) (\exp (i S / ℏ) (\vec{\nabla} \sqrt{P} + i / ℏ \sqrt{P} \exp (i S / ℏ) \vec{\nabla} S))) = \frac{ℏ}{m} ℑ (\sqrt{P} \vec{\nabla} \sqrt{P} + \frac{i}{ℏ} P \vec{\nabla} S) = \frac{P}{m} \vec{\nabla} S$ .

Example

$Ψ \sim \exp i \frac{\vec{p} \cdot \vec{x} - E t}{ℏ}$ , $\vec{j} = \frac{ℏ}{m} \frac{\vec{p}}{ℏ} = \frac{\vec{p}}{m} = \frac{P}{m} \vec{\nabla} S = \frac{1}{m} \vec{\nabla} S$ . So, $\vec{\nabla} S = \vec{P}$ .

Inserting into Schrodinger’s Equation - Hamilton-Jacobi

$Ψ (\vec{x}, t) = \sqrt{P (\vec{x}, t)} \exp (\frac{i}{ℏ} S (\vec{x}, t))$ , $P (\vec{x}, t) = | Ψ (\vec{x}, t) |^{2}$ .

$i ℏ \frac{\partial}{\partial t} Ψ = (- \frac{ℏ^{2}}{2 m} {\vec{\nabla}}^{2} + V) Ψ = i ℏ [\frac{\partial}{\partial t} \sqrt{P} + \frac{i}{ℏ} \sqrt{P} \frac{\partial S}{\partial t}] \exp \frac{i}{ℏ} S = - \frac{ℏ^{2}}{2 m} {\vec{\nabla}}^{2} (\sqrt{P} \exp \frac{i}{ℏ} S) + V \sqrt{P} \exp \frac{i}{ℏ} S$ .

$i ℏ [\frac{\partial}{\partial t} \sqrt{P} + \frac{i}{ℏ} \sqrt{P} \frac{\partial S}{\partial t}] = - \frac{ℏ^{2}}{2 m} [{\vec{\nabla}}^{2} (\sqrt{P}) + (\frac{i}{ℏ} \sqrt{P} {\vec{\nabla}}^{2} S + \frac{2 i}{ℏ} \vec{\nabla} (\sqrt{P} (\vec{\nabla} S) - \frac{1}{ℏ^{2}} \sqrt{P} | \vec{\nabla} S |^{2}))]$ .

In the classical limit, we see $0 = \sqrt{P} \frac{\partial S}{\partial t} + \frac{\sqrt{P}}{2 m} | \vec{\nabla} S |^{2} + V \sqrt{P}$ . So, $\frac{\partial S}{\partial t} + \frac{1}{2 m} | \vec{\nabla} S |^{2} + V = 0$ which is the Hamiltonian-Jacobi equation with $S$ the Hamilton’s principle function (action?).

Special Case

If $H$ is time independent then $S (\vec{x}, t) = W (\vec{x}) - E t$ . So, $Ψ (\vec{x}, t) = \sqrt{P} \exp \frac{i}{ℏ} (W (\vec{x}) - E t)$ , which gives a time independent and time dependent state. In HJ, $\frac{1}{2 m} | \vec{\nabla} W (\vec{x}) |^{2} + V (\vec{x}) = E$ . Thus, $| \vec{\nabla} W (\vec{x}) |^{2} = 2 m (E - V (\vec{x}))$ .

One Dimensional

$\frac{d}{d x} W = \pm \sqrt{2 m (E - V (x))}$ , $E > V (x)$ . So, $W (x) = \pm \int^{x} d x^{'} \sqrt{2 m (E - V (x^{'}))}$ .

Helper Info

${\vec{\nabla}}^{2} (\sqrt{P} \exp \frac{i}{ℏ} S) = {\vec{\nabla}}^{2} (\sqrt{P}) \exp \frac{i}{ℏ} S + (\frac{i}{ℏ} \sqrt{P} {\vec{\nabla}}^{2} S + \frac{2 i}{ℏ} \vec{\nabla} (\sqrt{P} (\vec{\nabla} S) - \frac{1}{ℏ^{2}} \sqrt{P} | \vec{\nabla} S |^{2})) \exp \frac{i}{ℏ} S$

Examples

Quantum Harmonic Oscillator

At the classical limit, we see that the time evolved uncertainty of position follows a localized mass exactly.