Semiclassical and quantum analysis of a free-particle Hermite wave function

In this Brief Report we discuss a solution of the free-particle Schrödinger equation in which the time and space dependence are not separable. The wave function is written as a product of exponential terms, Hermite polynomials, and a phase. The peaks in the wave function decelerate and then accelerate around $t=0$. We analyze this behavior within both a quantum and a semiclassical regime. We show that the acceleration does not represent true acceleration of the particle but can be related to the envelope function of the allowed classical paths. Comparison with other “accelerating” wave functions is also made. The analysis provides considerable insight into the meaning of the quantum wave function.

Figure 1

Contour map of the density associated with the wave function of Eq. (2) as a function of time and space. This was evaluated for $t_c=1\text{a.u.}$ with $n=2$. Superimposed on this are the hyperbolas given by Eq. (6).

Figure 2

Phase-space curves ($p$ vs $x$ for the family of paths described by Eqs. (9) and (10). At $t=0$ this is a circle in phase space which shears into an ellipse as time proceeds.

Figure 3

The family of allowed classical paths for the system described by Eq. (10). All quantities are in atomic units.