Description

The free particle wave packet program displays the time evolution of a free (V = 0 everywhere) initial Gaussian wave packet in position and momentum space. The default wave functions have expectation value of x = 0 and expectation value of p = 0, respectively. The Schrödinger equation determines the time evolution of this wave packet. Although the square of the area of the wave packet (total probability) remains constant, the wave packet spreads in position space as shown in this model of the analytic solution. Why doesn't the wave packet change in momentum space?

The time-dependent wave packet is complex but do we represent such complex functions? An arbitrary complex wave function can be written as:

Ψ(x,t) = ΨRe(x,t) + iΨIm(x,t)

and we plot real (red) and imaginary (blue) parts of the wave function when the Re/Im option is slected. We can also write the wave function as

Ψ(x,t) = A(x,t) e^iθ(x,t),

where the angle, θ(x,t) = atan[ΨIm(x,t)/ΨRe(x,t)] is the phase of the wave function. To visualize this wave function representation, the probability amplitude A(x,t) is the height and the phase is a color in the hue-saturation-brightness color space.