Interpretations of the Wavefunction
- Quantum Mechanics 1.13
from 「Modern Quantum Mechanics」: Sakurai, J. J.
The position and momentum operators.
Interpretations of the Wavefunction
The Copenhagen Interpretation
The Copenhagen interpretation is the most widely accepted interpretation of quantum mechanics. It was developed by Niels Bohr and Werner Heisenberg in the 1920s. The Copenhagen interpretation is a statistical interpretation of the wavefunction, which states that the square of the wavefunction gives the probability of finding a particle at a given position.
Probability density and Probability current
By the Copenhagen interpretation, the probability density is given by,
\[ \rho(\b{x},t) = \abs{\Psi(\b{x},t)}^2 \]
Similarly, the probability current of a spin-0 particle is given by,
\[ \b{j}(\b{x},t) = \frac{\hbar}{2mi} \left( \Psi^\ast\grad\Psi-\Psi\grad\Psi^\ast \right) \]
We can verify whether the probability continuity equation is satisfied, since the whole probability must be conserved.
\[ \begin{align*} \div\b{j} &= \frac{\hbar}{2mi} \left( \Psi^\ast\grad^2\Psi-\Psi\grad^2\Psi^\ast \right) \nl &= -\frac{\hbar}{2mi} \frac{2m}{\hbar^2} \left[ \Psi^\ast(\hat{H}-V)\Psi - \Psi(\hat{H}-V)\Psi^\ast \right] \nl &= \frac{i}{\hbar} \left[ \Psi^\ast\hat{H}\Psi - \Psi\hat{H}\Psi^\ast \right] \nl &= - \left[ \Psi^\ast \pdv{ }{t}\Psi - \Psi \pdv{ }{t}\Psi^\ast \right] \nl &= - \pdv{ }{t} \abs{\Psi}^2 \nl &= - \pdv{\rho}{t} \end{align*} \]
Therefore, the probability continuity equation is satisfied.
\[ \pdv{\rho}{t} + \div\b{j} = 0 \]
Classical limit
To understand the physical significance of the wavefunction, let’s write as
\[ \Psi(\b{x},t) = \sqrt{\rho(\b{x},t)} \exp\left[ \frac{iS(\b{x},t)}{\hbar} \right] \]
We can then write the probability current as
\[ \b{j} = \frac{\rho\grad S}{m} \]
Since classically $\b{j}=\rho\b{v}$, we can assume that $\grad S/m=\b{v}$. This yields,
\[ \grad S = \b{p} \]
To explore the classical limit of the wavefunction, let’s also substitute the wavefunction above into the Schrödinger equation. After some complicated differentiations, we get
\[ -\frac{\hbar^2}{2m} \left[ \laplacian\sqrt{\rho} + \frac{2i}{\hbar} (\grad\sqrt{\rho})\cdot(\grad S)-\frac{1}{\hbar^2} \sqrt{\rho} \abs{\grad S}^2 +\frac{i}{\hbar} \sqrt{\rho} \laplacian S \right] \exp\left[ \frac{iS(\b{x},t)}{\hbar} \right] + V \sqrt{\rho} \exp\left[ \frac{iS(\b{x},t)}{\hbar} \right] = i\hbar \left[ \pdv{\sqrt{\rho}}{t}+\frac{i}{\hbar}\sqrt{\rho}\pdv{S}{t} \right] \exp\left[ \frac{iS(\b{x},t)}{\hbar} \right] \]
Assume that $\; \hbar\laplacian S\ll \abs{\grad S}^2 $, we can simply the equation.
\[ -\frac{\hbar^2}{2m} \laplacian\sqrt{\rho} - \frac{i\hbar}{m} (\grad\sqrt{\rho})\cdot(\grad S) + \frac{1}{2m} \sqrt{\rho} \abs{\grad S}^2 + V \sqrt{\rho} = i\hbar \pdv{\sqrt{\rho}}{t} - \sqrt{\rho}\pdv{S}{t} \]
We can then collect terms that do not explicitly contain $\hbar$ to obtain a nonlinear PDE for $S$.
\[ \frac{1}{2m}\abs{\grad S(\b{x},t)}^2 + V(\b{x}) + \pdv{S(\b{x},t)}{t} = 0 \]
We recognize this to be the Hamilton-Jacobi equation in classical mechanics, where $S(\b{x},t)$ stands for Hamiltonian’s principal function. The interpretation for the fact is that, classical mechanics is contained in quantum mechanics in the $\hbar\to0$ limit.
