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.

 

[paranormal]

I would recommend using numerical methods to solve this integral. One possible approach is to discretize the integral and use numerical integration techniques such as Simpson's rule or the trapezoidal rule. Alternatively, you could also try using a computer program specifically designed for solving integrals, such as Wolfram Alpha or Mathematica. Additionally, if the integral cannot be solved analytically, you could try approximating it using a series expansion or using other methods such as Monte Carlo simulation. It is important to keep in mind that sometimes, equations in physics cannot be solved analytically and numerical methods must be used instead.