4.1 THE FREE PARTICLE

The case of a free particle moving through space is a useful first study, even though some of the later examples are intrinsically more interesting. Without further ado, here is a movie showing the free particle wavepacket propagating:
  1. QT Movie 4.1.2: Propagation of Free Particle - High Resolution (930kbytes).
The important points of physics to note are the packet velocity, the broadening, and the consequent reduction in amplitude:

Wavepacket Velocity. The dispersion relation for plane wave solutions to Schroedinger's equation in the units used here ( m=1/2, h bar=1) can be used to determine the mean wavepacket group velocity:
 
 

Eqtn 4.1.1

Suggestion: Try checking this prediction against a direct measurement from the movie.

Broadening. Standard quantum mechanics texts (eg. Merzbacher) show the dispersion relation for plane waves also means that a wavepacket will necessarily broaden as time evolves. Specifically the packet width (in the units used here) becomes,
 
 

Eqtn 4.1.2

This broadening can be clearly seen in the movie, but a quantitative comparison with theory is probably done more easily by measuring the consequent reduction in height, as explained below.

Amplitude Decrease. The total integrated probability of finding the particle must remain constant, and so if the width of the gaussian wavepacket increases, the height must decrease, and by the same factor. The height of the probability packet psi psi* after time t is therefore given by,
 
 

Eqtn 4.1.3

Suggestion: Measure the height from the final frame and compare with theory.

Reference: "Quantum Mechanics", E. Merzbacher, Wiley International, New York, 1970, 2nd edition, Exercise 8.12

Comments etc to : jo@nat.vu.nl