Classical Limits of Quantum Mechanics
\(\hbar\to0\).
| Classical | Quantum |
| \(\frac{d A}{dt}=[A,H]_{classical}\) | \(\frac{d A_H}{dt}=\frac{i}{\hbar}[H,A_H]\) |
Where \([A,H]_{classical}\) 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: \([]_{classical}\Leftrightarrow[]_{QM}/(i\hbar)\). 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 |
\(\Psi(\vec{x},t) = \sqrt{\mathcal{P}(\vec{x},t)}\exp\left(\frac{i}{\hbar}S(\vec{x},t)\right)\), \(\mathcal{P}(\vec{x},t)=|\Psi(\vec{x},t)|^2\).
\(\vec{j}(\vec{x},t)=\frac{\hbar}{m}\Im(\Psi^*\vec{\nabla}\Psi) = \frac{\hbar}{m}\Im(\sqrt{\mathcal{P}}\exp(-iS/\hbar)(\exp(iS/\hbar)(\vec{\nabla}\sqrt{\mathcal{P}}+i/\hbar \sqrt{\mathcal{P}}\exp(iS/\hbar)\vec{\nabla}S))) = \frac{\hbar}{m}\Im(\sqrt{\mathcal{P}}\vec{\nabla}\sqrt{\mathcal{P}} + \frac{i}{\hbar}\mathcal{P}\vec{\nabla}S) = \frac{\mathcal{P}}{m}\vec{\nabla}S\).
Example
\(\Psi \sim \exp i\frac{\vec{p}\cdot\vec{x}-Et}{\hbar}\), \(\vec{j}=\frac{\hbar}{m}\frac{\vec{p}}{\hbar}=\frac{\vec{p}}{m} = \frac{\mathcal{P}}{m}\vec{\nabla}S= \frac{1}{m}\vec{\nabla}S\). So, \(\vec{\nabla}S=\vec{P}\).
Inserting into Schrodinger’s Equation - Hamilton-Jacobi
\(\Psi(\vec{x},t) = \sqrt{\mathcal{P}(\vec{x},t)}\exp\left(\frac{i}{\hbar}S(\vec{x},t)\right)\), \(\mathcal{P}(\vec{x},t)=|\Psi(\vec{x},t)|^2\).
\(i\hbar\frac{\partial}{\partial t}\Psi=\left(-\frac{\hbar^2}{2m}\vec{\nabla}^2+V\right)\Psi = i\hbar\left[\frac{\partial}{\partial t}\sqrt{\mathcal{P}}+\frac{i}{\hbar}\sqrt{\mathcal{P}}\frac{\partial S}{\partial t}\right]\exp\frac{i}{\hbar}S = -\frac{\hbar^2}{2m}\vec{\nabla}^2(\sqrt{\mathcal{P}}\exp\frac{i}{\hbar}S)+V\sqrt{\mathcal{P}}\exp\frac{i}{\hbar}S\).
\(i\hbar\left[\frac{\partial}{\partial t}\sqrt{\mathcal{P}}+\frac{i}{\hbar}\sqrt{\mathcal{P}}\frac{\partial S}{\partial t}\right] =-\frac{\hbar^2}{2m}\left[\vec{\nabla}^2(\sqrt{\mathcal{P}}) + \left(\frac{i}{\hbar}\sqrt{\mathcal{P}}\vec{\nabla}^2S+\frac{2i}{\hbar}\vec{\nabla}(\sqrt{\mathcal{P}}(\vec{\nabla}S) - \frac{1}{\hbar^2}\sqrt{\mathcal{P}}|\vec{\nabla}S|^2)\right)\right]\).
In the classical limit, we see \(0=\sqrt{\mathcal{P}}\frac{\partial S}{\partial t} + \frac{\sqrt{\mathcal{P}}}{2m}|\vec{\nabla}S|^2 + V\sqrt{\mathcal{P}}\). So, \(\frac{\partial S}{\partial t} + \frac{1}{2m}|\vec{\nabla}S|^2 + V=0\) which is the Hamiltonian-Jacobi equation with \(S\) the Hamilton’s principle function (action?).
Special Case
If \(H\) is time independent then \(S(\vec{x},t)=W(\vec{x})-Et\). So, \(\Psi(\vec{x},t)=\sqrt{\mathcal{P}}\exp\frac{i}{\hbar}(W(\vec{x})-Et)\), which gives a time independent and time dependent state. In HJ, \(\frac{1}{2m}|\vec{\nabla}W(\vec{x})|^2 + V(\vec{x}) = E\). Thus, \(|\vec{\nabla}W(\vec{x})|^2 = 2m(E-V(\vec{x}))\).
One Dimensional
\(\frac{d}{dx}W = \pm\sqrt{2m(E-V(x))}\), \(E>V(x)\). So, \(W(x)=\pm\int^x dx'\sqrt{2m(E-V(x'))}\).
Helper Info
\(\vec{\nabla}^2(\sqrt{\mathcal{P}}\exp\frac{i}{\hbar}S) = \vec{\nabla}^2(\sqrt{\mathcal{P}})\exp\frac{i}{\hbar}S + \left(\frac{i}{\hbar}\sqrt{\mathcal{P}}\vec{\nabla}^2S+\frac{2i}{\hbar}\vec{\nabla}(\sqrt{\mathcal{P}}(\vec{\nabla}S) - \frac{1}{\hbar^2}\sqrt{\mathcal{P}}|\vec{\nabla}S|^2)\right)\exp\frac{i}{\hbar}S\)
Examples
- Quantum Harmonic Oscillator
At the classical limit, we see that the time evolved uncertainty of position follows a localized mass exactly.