# Wave packet with increasing time

1. Aug 25, 2014

### mcheung4

I am trying to understand how a gaussian packet varies in time.

Suppose we have a Gaussian wave packet that is displaced form the origin by an amount x0 and given initial momentum p0. So the wave function in coordinate space is

ψ(x,0) =$\frac{√β}{√√\pi}$exp(-β2(x-x0)2/2)*exp(ip0x/ħ)

where β is some constant such that ψ does not correspond to the ground state of the harmonic oscillator (not an eigenstate).

Now given Δx0 = 1/(β√2), we can eliminate β and find ψ2

ψ(x,0)2 = 1/√2$\pi$ * 1/Δx0 * exp(-(x-x0)2/2(Δx0)2)

Then

ψ(x,t)2 = 1/√2$\pi$ * 1/Δx(t) * exp(-(x-x0)2/2(Δx(t))2)

Q1: why does Δx0 = 1/(β√2)? Isn't Δx0 a definite value so it should not have any deviation?
Q2: How does ψ(x,0)2 transform to ψ(x,t)2 and then we have Δx(t) replacing Δx0?

2. Aug 25, 2014