Semiclassical and Quantum Analysis of a Focussing Free Particle Hermite Wavefunction

In this paper we discuss a solution of the free particle Schroodinger equation in which the time and space dependence are not separable. The wavefunction is written as a product of exponential terms, Hermite polynomials and a phase. The peaks in the wavefunction decelerate and then accelerate around t = 0. We analyse this behaviour within both a quantum and a semi-classical 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" wavefunctions is also made. The analysis provides considerable insight into the meaning of the quantum wavefunction.