PHY251 Gaussians in Quantum Mechanics

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 $\pi$, and you may have wondered where it came from. This is directly tied to the special case $G(1)$. The constant $\pi$ 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 $(\theta,r)$ 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 $a>0$ 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 $x^n$ as $x\cdot x^{n-1}$ and integrating by parts, we get :

Repeating this process, we obtain :

For the purposes of the problem sheet, we can mention the case of $n$ even. Writing $n=2k$, with $k\in\mathbb{N}$ :

where $\Gamma$ is the well-known Gamma function, for which a closed form exists ($k\in\mathbb{N}$) :

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 $n=0$ separates into two distinct, Gaussian modes as $n$ increases, with parity $(-1)^n$. Play around with Mathematica; how does adding polynomial terms modify these modes?

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