7. Correspondence Principle
Reading time: ~30 minutes | Pages: 6
Quantum mechanics must reduce to classical mechanics in appropriate limits.
Statement
In the limit of large quantum numbers ($n \gg 1$) or small $\hbar$, quantum predictions approach classical results.
Ehrenfest's Theorem
Expectation values obey classical-like equations:
$$\frac{d\langle\hat{x}\rangle}{dt} = \frac{\langle\hat{p}\rangle}{m}, \quad \frac{d\langle\hat{p}\rangle}{dt} = -\left\langle\frac{\partial V}{\partial x}\right\rangle$$
WKB Approximation
Semiclassical regime ($\lambda \ll$ characteristic length):
$$\psi(x) \approx \frac{C}{\sqrt{p(x)}}e^{i S(x)/\hbar}$$
where $S(x) = \int p(x')dx'$ is classical action
Classical Limit Conditions
- Large quantum numbers: $n \gg 1$
- Large action: $S \gg \hbar$
- Short wavelength: $\lambda \ll L$ (system size)
- Weak quantum fluctuations: $\Delta x \cdot \Delta p \gg \hbar$
Example: Harmonic Oscillator
For large $n$, quantum distribution approaches classical:
$$|\psi_n(x)|^2 \xrightarrow{n\to\infty} \frac{1}{\pi\sqrt{A^2 - x^2}} \quad \text{(classical PDF)}$$