Physics 4410, Spring '99 Homework #10

Hints:

In any of these problems, try not to reinvent the wheel. If Griffiths (or my notes) has done an integral or found a normalization constant already for you, feel free to just use the result! (Quote an equation number if it comes from Griffiths, a page number if it comes from my notes)

1) Unit analysis for the true ground state energy should not (can not!) involve "b", which is merely a variational parameter. It can only involve alpha, or hbar, or m, to various powers.

However, <V> can involve "b", because the hidden state in the bra and ket is the trial state, which explicit involves "b".

2) Griffiths Example 3 does some important work for you, see the comment at the beginning of these hints!

Griff. 2.111 gives the exact answer. How did you do?

3) The angular part of the wave function I've picked is basically guaranteed to be correct for the ground state, so I haven't put any "variable parameters" in it. (All angular integrals you have to do for this problem should be totally trivial, just a bunch of 4 Pi's...)

The exponential radial function I picked is exactly right for hydrogen's ground state, so it seems like an o.k. guess for a 3-D harmonic oscillator. (But, it's not perfect!)

The normalization constant N is NOT a variational parameter - you'll have to fix it to ensure normalization. Be careful - remember this is a 3-D wave function now... (Don't forget, e.g. the usual factor of r^2 in radial integrals...)

Note: Griffiths Eq. 4.13 tells you the Laplacian, if you've forgotten it.

A handy integral to know throughout this problem is

A useful intermediate result to check yourself: I get .

(Here you must show your work, of course - no fair just quoting THIS result!)

4) My notes (PP. 166-168, especially) should be much more helpful than Griffiths! That's because I kept calling the original nuclear charge "Z", but Griffiths started plugging in the number "2" right away, so you can't clearly see what the dependence on Z is anymore in his equations. (His "Z" is my "Z*" in the notes, the effective charge....)

Extra credit problem hint: Evaluate with a convenient trial wave function. and show that this expectation value can always be made negative. (One possibility is e.g. N Exp[-beta^2 x^2], although I can think of other ways to solve the problem)