# Classical waves have a quantum correspondence only when $E >V$

In the book Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer Series in Materials Science) one reads the following:

For waves of a given energy or frequency, the quantum and classical wave equations can be put in the same form $$\nabla^2\phi + \kappa^2\phi = 0,\tag{2.8}$$ where $$\kappa^2 = \frac{2m(E - V)}{\hbar^2}\tag{2.9}$$ for the quantum case, and $$\kappa^2 = \frac{\omega^2}{v^2}\tag{2.10}$$ for the classical case. Here a striking difference between the quantum wave and the classical waves is already apparent, because whereas $E - V$ can be negative in (2.9), $\omega^2/v^2$ is always positive. Therefor, classical waves have a quantum correspondence only when $E > V$.

It seems to me that instead of saying that classical waves have a quantum correspondence only when $E >V$ one should say that quantum waves have a classical correspondence only when $E >V$, since there are quantum waves when $\kappa^2 <0$ but there can't be classical waves for those values of the parameter $\kappa$.

Am I missing something here?

Your equation 2.8 is actually the Helmholtz equation which you get from different partial differential equations involving time by separating out the time dependence by the method of separation of variables yielding the separation constant $\kappa^2$, which can be any number. In the time independent Schrödinger equation, equation 2.8 + 2.9, for $E <V$ you get a negative $\kappa^2$ so that $\kappa$ is imaginary. This means that you have an exponentially damped wave for a total particle energy $E$ smaller than its potential energy $V$, a situation which cannot occur in classical physics. There can also be "classical waves" with $\kappa^2 <0$, i.e., exponentially damped waves. For example electromagnetic waves in waveguides below the cut-off frequency are purely damped waves. Also light waves in the case of total reflection have exponential damping.