The Method of Stationary Phase: Location of the packet

Figure 4 gives a representation of the integrand in (5). The integrand at each value of k, , is sketched as a complex vector with its base placed at the corresponding point k along the real axis. The significant contributions to the integral all come from the points near where the amplitude of the numbers is most significant. The figure shows two cases. In one case (the solid arrows), the phases vary rapidly across the region of significant contributions, whereas in the second case (the dashed arrows), the phases vary slowly in the region of significance.

The value of the integral is the sum of these vector obtained by adding them tail-to-head as in Figure 5. In the case where the phases vary rapidly near , relatively little net progress is made away from the origin of the complex plane as the contributing vectors spin around. The result of this behavior is a small magnitude for the wave function, . In the second case where the phases vary slowly, on the other hand, the vectors all line up to produce a large final magnitude for the wave function .

  
Figure 5: Total integral analyzed using stationary phase

We see, therefore, that the largest come when the phase is as constant, as stationary, as possible in the region where is most significant. Defining as the total phase of the integrand, this is the condition that

  

Applying the general condition (6) to locate the regions of greatest contribution from an integral is referred to as The Method of Stationary Phase.

In our specific case, the great magnitude of occurs when (7) is satisfied, when

The most probable location for the particle follows the same trajectory in time as does a classical particle traveling with the velocity . The physical interpretation of the states near as beams of particles traveling with the classically expected velocity is indeed correct!

The analysis here further gives the location of the packet at time t=0,

and the time when the center of the packet crosses the origin,

The three lessons of general applicability to be taken from this section are

1. Beams of particles carry wave packets at the classically expected velocity.
2. The Method of Stationary Phase tells us that the most significant contributions from the integral of a complex integrand come when the phase of the integrand does not vary at the point of its maximum magnitude.
3. The initial location of a wave packet and the time when it crosses x=0 are both determined by the derivative of the phase of the weights of superposition evaluated at the momentum where the packet is concentrated.