The objective of this note is to derive the wave function for a wave packet if time is large.
To shorten the writing, the Fourier integral (7.64) for
will be abbreviated as:
The so-called “method of stationary phase” says that the integral is negligibly small as long as there are no stationary points 0 in the range of integration. Physically that means that the wave function is zero at large time positions that cannot be reached with any group velocity within the range of the packet. It therefore implies that the wave packet propagates with the group velocity, within the variation that it has.
To see why the integral is negligible if there are no stationary
points, just integrate by parts:
For large time positions with values within the range of packet group velocities, there will be a stationary point to . The wave number at the stationary point will be indicated by , and the value of and its second derivative by and . (Note that the second derivative is minus the first derivative of the group velocity, and will be assumed to be nonzero in the analysis. If it would be zero, nontrivial modifications would be needed.)
Now split the exponential in the integral into two,
Now split function apart as in
That leaves the first part, , which
Fresnel integralthat can be looked up in a table book. Away from the edges of the wave packet, the integration range can be taken as all , and then
Right at the edges of the wave packet, modified integration limits for must be used, and the result above is not valid. In particular it can be seen that the wave packet spreads out a distance of order beyond the stated wave packet range; however, for large times is small compared to the size of the wave packet, which is proportional to .
For the mathematically picky: the treatment above assumes that the wave packet momentum range is not small in an asymptotic sense, (i.e. it does not go to zero when becomes infinite.) It is just small in the sense that the group velocity must be monotonous. However, Kaplun’s extension theorem implies that the packet size can be allowed to become zero at least slowly. And the analysis is readily adjusted for faster convergence towards zero in any case.