Since this week’s QM problem sheets involves Gaussian integrals, I thought it might be a good idea to derive a few results of interest, rather than just stating them. Let’s then define the generic Gaussian integral:

When looking at the wavefunctions of the quantum harmonic oscillators, you may have noticed a recurring factor of , and you may have wondered where it came from. This is directly tied to the special case . The constant has of course deep connections with geometry, and it’s by making a change of coordinates to work on a disk that we will show how it comes about. Of course, one variable is not enough to get to the system we’re after, so let’s instead look at the product of two such Gaussian integrals:

Now that we’ve isolated this special case, we can work out the generic expression for any real through a simple change of variable in the measure:

Modified Gaussian integrals

It’s interesting, especially when working with the wavefunctions of excited states of the harmonic oscillator, to consider also the action of a polynomial on a Gaussian. You can scroll down to the Mathematica section to see how this relates to the parity and localisation of the wavefunctions. Let’s then define:

Decomposing as and integrating by parts, we get:

Repeating this process, we obtain:

For the purposes of the problem sheet, we can mention the case of even. Writing , with :

where is the well-known Gamma function, for which a closed form exists ():

Having said all this, let’s now compute one of the integrals that turns up in the problem sheet:

A bit of Mathematica…

To see the effects of including a polynomial (here we’ll keep it to a simple power law) factor to the Gaussian, you can use the Manipulate method in Mathematica as follows:

Manipulate[Plot[x^n*Exp[-x^2],{x,-5,-5}],{n,0,10,1}]

You should be able to see how the “single” wavefunction at separates into two distinct, Gaussian modes as increases, with parity . Play around with Mathematica; how does adding polynomial terms modify these modes?