Mandelung and Van Vleck

The Schrödinger equation describes the time evolution of quantum states:

\[ H(t) \left| \psi , \, t \right\rangle = i \hbar \frac{\partial}{\partial t} \left| \psi , \, t \right\rangle \]

Expanding the state in the coordinate basis
\[ \left| \psi, \, t \right\rangle = \int dx \, \psi(x, \, t) \left| x \right\rangle \]
allows us to write the Schrödinger equation for the wavefunction $ \psi(x, \, t) $. Since we are working with vectors in a Hilbert space, the wavefunction is a complex number. One could do three things. First one could write the complex wavefunction in terms of real and imaginary parts:

\[ \psi = \psi_{1} + i \psi_{2} \]

Second, one could write the complex wavefunction in terms of its magnitude and phase:

\[ \psi = \sqrt{\rho} \exp{(i \theta)} \]

And third, one could shut-up and work with the complex wavefunction like everyone else does. It turns out that writing the wavefunction in terms of its magnitude and phase is known as the Mandelung ansatz. This is related to what one does in the Van Vleck approximation where one writes the phase of the wavefunction as

\[ \theta = -\frac{1}{\hbar} \Sigma \]

and looks at the limit $\hbar \rightarrow 0$. The Schrödinger equation becomes the Hamilton-Jacobi equation with a current continuity equation:

\[ H_{cl} + \frac{\partial \Sigma}{\partial t} = 0 \qquad \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial q} \left( \rho \frac{\partial H_{cl}}{\partial p} \right) = 0 \]

This is the semiclassical limit of the Mandelung equations.