How Can You Find ψ(x,t) from ψ(x) for a Free Particle?

[JordanGo]

Homework Statement

Consider the free-particle wavefunction,
ψ(x)=(pi/a)^(-1/4)*exp(-ax^2/2)

Find ψ(x,t)

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?

 

[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.

 

[jmcelve]

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

 

[vela]

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.