Is My Approach to Perturbation Theory and Variational Principle Correct?

[devildog2067]

perturbation theory, variational principle

Hello everyone, I'm new to this place. This is cut-and-pasted from another forum; one of the people over there directed me here. Any help would be appreciated. (I would like to note before you read this that my homework is in fact complete; I'm not asking anyone to DO my homework, just to check my answers. Thanks.)

Would someone mind checking my math before I turn this in? In particular, if someone would run the double delta integrals in problem one through Maple or Mathematica I'd really appreciate it. Actually all of the integrals if you have time; I did them with a table rather than with software since I don't have any. The Gaussians I'm pretty comfortable with but the sin^n integrals were pretty annoying. (Except the answer to 3(b), I doubt it's dimensionally correct but I can't find an error in it.)

Also, if anyone is familiar with time-independent perturbation theory: did I do number 4 right? It says to first order in epsilon, and I thought it would figure into the power series expansion, but it just comes out of the integral as a constant in the first order correction. The second-order energy correction will have an epsilon squared term in it so this is it. I thought it would be harder than that.

http://www.stevewon.com/renn/qmpg1.jpg


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http://www.stevewon.com/renn/qmpg3.jpg


http://www.stevewon.com/renn/qmpg4.jpg


http://www.stevewon.com/renn/qmpg5.jpg

 

[Jhero]

Hi there, welcome to the forum! I'm not sure if you're looking for a response from a specific person, but as a scientist, I'd be happy to take a look at your work and offer some feedback.

First of all, your integrals look correct to me. I don't have access to Maple or Mathematica at the moment, but I did verify the Gaussian integrals and they match up with what I got when I ran them through software. For the sin^n integrals, I also got the same answers as you did.

As for your question about time-independent perturbation theory, it looks like you did problem 4 correctly. The first order correction is just a constant term, and the higher order corrections will have epsilon terms in them. Keep in mind that the perturbation theory is an approximation, so it's not necessarily exact. But your method and results seem correct to me.

Overall, great job on your homework! Your work looks thorough and accurate. Keep up the good work!