How to Change Variables for Integration in Quantum Mechanics?

[raintrek]

[SOLVED] Integration change of variables

Homework Statement

An electron is confined in an infinite one dimensional well where 0 < x < L with L = 2 x 10^-10m. Use lowest order perturbation theory to determine the shift in the third level due to the perturbation:

$$V(x) = V_{0}\left(\frac{x}{L}\right)^{2}$$

where $$V_{0} = 0.01eV$$.

After a change of variables, the following integral will be useful:

$$\int^{3\pi}_{0}\phi^{2}sin^{2}\phi d\phi = \frac{9}{2}\pi^{3} - \frac{3}{4}\pi$$

The Attempt at a Solution

I've evaluated this question into the following integral:

$$\Delta E_{3}^{(3)} = \frac{2}{L}\frac{V_{0}}{L^{2}}\int^{L}_{0} x^{2} sin^{2}\left(\frac{3\pi x}{L}\right) dx$$

However I have no idea how to "change variables" with an integral like this, let alone how to get the limits change from 0-L to 0-3pi. Can anyone offer assistance? Many thanks in advance...

 

[nicksauce]

I believe the change of variables they use is

$$\frac{3\pi x}{L} \rightarrow \phi$$

So clearly as $$x \rightarrow L$$ , $$\phi \rightarrow 3\pi$$

 

[raintrek]

hmm, if that was the case, wouldn't the equation take the form

$$\Delta E_{3}^{(3)} = \frac{2}{L}\frac{V_0}{L^2}\frac{L^{3}}{(3\pi)^{3}}\int^{3\pi}_{0}\phi^{2}sin^{2}\phi d\phi$$

(Note the L^3/... prefactor) ??

 

[nicksauce]

So then can't you plug in for
$$\int^{3\pi}_0 \phi^2 \sin^2{\phi}d\phi$$
which you are given? I don't see the problem.

 

[raintrek]

^ I can, I was just seeking assurance that I'd calculated that L^3 prefactor correctly after substitution...

 

[nicksauce]

It all looks correct... I know nothing about perturbation theory, so I can't say if the answer qualitatively makes sense with no L dependence.

 

[raintrek]

Yep, that works good, thanks for your help nicksauce!