Answer to Question #167703 in Quantum Mechanics for Sherine Horo

Question

Show that quantum mechanics reduces to classical mechanics for 𝒉 →0

Answer 1

The wave function of a classical system should have a form $\Psi = a e^{i\phi}$ with $a$ a function that is changing very slowly (as classical mechanics are deterministic), the phase function is related to the action in the classical case as $S=\phi\cdot \hbar$ by analogy with the classical optics (Fermat's principe). Now by inserting this form of $\Psi$ in the Schrodinger's equation we get

$a \partial_t S-i\hbar \partial_t a + \frac{a}{2m} (\nabla S)^2 - \frac{i\hbar}{2m}a\Delta S-\frac{i\hbar}{m}\nabla S \nabla a - \frac{\hbar}{2m}\Delta a + Va=0$

Now by writing separate equations for the real and imaginary parts (as a and S are real) we get:

$\partial_t S + \frac{1}{2m}(\nabla S)^2+V-\frac{\hbar^2}{2ma} \Delta a=0$

$\partial_t a + \frac{a}{2m} \Delta S + \frac{1}{m} \nabla S \nabla a=0$

The first equation in the limit $\hbar \to 0$ gives us the Hamilton-Jacobi equation that describes the classical mechanics.

The second equation defines classical velocity in the terms of quantum mechanics.

The reference of these calculations is Landau, Lifshitz Quantum Mechanics: Non-Relativistic Theory. Vol. 3.