# Finding psi(x,t) using psi(x)

1. Sep 20, 2012

### JordanGo

1. The problem statement, all variables and given/known data
Consider the free-particle wavefunction,
ψ(x)=(pi/a)^(-1/4)*exp(-ax^2/2)

Find ψ(x,t)

3. The attempt at a solution

The wavefunction is already normalized, so the next thing to find is coefficient expansion function (θ(k)), where:

θ(k)=∫dx*ψ(x)*exp(-ikx) from -infinity to infinity

But this equation seems to be impossible to solve without error function (as maple 16 tells me).

Is there any trick to solve this?

2. Sep 20, 2012

### voko

The error function has very nice properties at the infinity, so you should be able to compute the integral. Alternatively, you could use the apparatus of complex analysis to evaluate the integral.

3. Sep 21, 2012

### jmcelve

Why not use some of the simple Gaussian integral tricks (like completing the square in the exponential)? Or am I missing something?

4. Sep 22, 2012

### vela

Staff Emeritus
You can avoid the error function because of the limits on the integral, or equivalently, use the nice properties of the error function at those limits.

You should really learn to crank this integral out by hand, though. The techniques used are useful to know.