1) Show your work, since the answer is given. (Where, e.g., does the 1/Sqrt[L] out front come from? Why is the energy quantized? Remember, this is not exactly the same as a particle in a box - the wave function is not constrained to go to zero anywhere in particular. What is the relevant boundary condition in this problem?)

Parts b and c do involve a fair bit of algebra.

2) Please note that the delta function arguments for x, y, and z are all located in different spots. You'll have to find the eigenvalues of a 3x3 matrix for this one, but if you did it right the matrix should be pretty simple...I only asked for the linear combo of the level that shifts upward, so you don't have to find all the eigenvectors!

3) For the eigenvalues of H, you can use MMA, although it's not all that hard to find them directly. In fact, this matrix is "block diagonal", which makes life even easier -you just need to find the eigenvalues of the non-zero square "block" submatrices. The exact and approximate answers should (will!) match up, but do realize you can't expect first order perturbation theory to generate any second order corrections!

4) Griffiths problems 5.1 and 5.2 tells you what the difference is between (the unperturbed part of) this problem and the pure hydrogen atom. (Not much!!) So, formulas from back in Griffiths chapter 4.2 all continue to hold, with minor corrections for the reduced mass.. I never said the mass of the particles is equal to the electron mass, so just keep it general. This problem, in fact, is a model used to describe the spectrum of heavy mesons, also called "quarkonium", and works quite nicely!

For part ii, think of the "handy theorem". (You might need to use some results from Griffiths chapter 4.3.2 to really firm up the argument.)