Time evolution of a free wave packet

Let

and assume that |g(k)| is centered at k₀ and has width Dk.

Let

  .

Then

 .

If we again use the use a Taylor series expansion,

,

then we already know that y(x,t) peaks at

.

The peak of the wave packet moves with the group velocity   v_g=dw/dk |_k=ko.

The velocity of a plane wave exp(i(kx-wt) is v_p=w/k and is called the phase velocity.  We have

.

A wave for which Dx<<x and Dp<<p can represent a classical particle moving with velocity v=v_g.

Spreading of a wave packet

We can terminate the Taylor series expansion

after the first term if all higher order terms are much smaller than 2p.  However, as t increases, the higher order terms increase steadily with t.  Therefore the second order term must be included for some t>t₁ .

We have

 ,   .

The wave packet is the of the form

.

   with   .

Now |x-x₀-vt|>>Dk^-1 is no longer sufficient for |t|<<Dk.  We need |x-x₀-vt-(k-k₀)x₁|>>Dk^-1 for the integrand to oscillate rapidly and the contribution to the integral to be zero.  The width of the wave packet increases with time.

Example:

(To explore the evolution of a Gaussian wave packet, see complement G_I, Cohen-Tannuoudji.  For a Gaussian wave packet the calculations can be carried out exactly, and no series expansion is necessary.)

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Summary

The wave function of a free particle is a wave packet.  The center of this wave packet moves with constant velocity v_g, called the group velocity.  The width of the wave packet increases with time, the wave packet spreads, we gradually loose position information.

Write your own program

This link shows you how to find a numerical solution of the time-dependent Schroedinger equation in one dimension for a wave packet representing an electron confined to a region between x=0 and x=L and presents you with an example program.

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