midterm_solutions

# midterm_solutions - PH 216 Midterm 2011 Instructions Your...

This preview shows pages 1–3. Sign up to view the full content.

PH 216, Midterm, 2011 Instructions: Your solutions are due Saturday at 9 AM , via email or in my office (I should be present on Saturday morning, but if you want to drop it friday night, or miss me, you can slip it under the door.) Late midterms will not be accepted except in case of natural catastrophe or national emergency. Feel free to consult any written (paper and ink) sources, but not computers or other people. I’ve made every effort to keep ambiguity out of the questions, but if some remains for you, I will try to remove it; please email ([email protected]) or feel free to call (325-6832 between 8 am and 9 pm). 1. An electron moves in Coulomb field centered at the origin of coordinates. With neglect of spin and relativistic corrections, the first excited state ( n = 2) is 4-fold degenerate: l = 0 , m = 0; l = 1 , m = ± 1 , 0. Consider what happens to this level in the presence of an additional non-central potentail V 0 = f ( r ) xy , where f ( r ) is some central function, well-behaved but not otherwise specified (it falls off rapidly enough as r → ∞ .) This perturbation is to be treated to first order. To this order the degenerate n = 2 level splits into several levels of different energies, each characterized by an energy shift Δ E and a degeneracy (perhaps non-degenerate, perhaps multiply degenerate.) (a) How many distinct energy levels are there? (b) What is the degeneracy of each? (c) Given the energy shift, call it A ( A > 0), for one of the levels, what are the values of the shifts for all of the other levels? Solution: With V = f ( r ) xy = f ( r ) r 2 sin 2 θ sin φ cos φ treated as perturbation, the unperturbed wave functions for energy level n = 2 are = 0 , m = 0 , R 20 ( r ) Y 00 , = 1 , m = 1 , R 21 ( r ) Y 11 , = 1 , m = 0 , R 21 ( r ) Y 10 , = 1 , m = - 1 , R 21 ( r ) Y 1 , - 1 . As they all correspond to the same energy, i.e. degeneracy occurs, we have to first calculate H 0 0 m 0 ‘m = h 0 m 0 | V | ‘m i = Z R 2 0 ( r ) R 2 ( r ) r 2 f ( r ) Y * 0 m 0 sin 2 θ sin φ cos φY ‘m dV. The required spherical harmonics are 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Y 00 = 1 4 π 1 2 , Y 11 = 3 8 π 1 2 sin θe , Y 10 = 3 4 π 1 2 cos θ, Y 1 , - 1 = 3 8 π 1 2 sin θe - . Considering the factor involving φ in the matrix elements H 0 0 m 0 ‘m we note that all such elements have on of the following factors: 2 π Z 0 sin φ cos φdφ = 0 , 2 π Z 0 e ± sin φ cos φdφ = 0 , except H 0 1 , - 1 , 1 , 1 and H 0 1 , 1 , 1 , - 1 , which have nonzero values H 0 1 , - 1 , 1 , 1 = 3 8 π Z [ R 21 ( r )] 2 r 4 f ( r ) dr π Z 0 sin 5 θdθ 2 π Z 0 sin φ cos φe - 2 = iA, H 0 1 , 1 , 1 , - 1 = - iA, with A = 1 5 Z [ R ( r )] 2 r 4 f ( r ) dr. We then calculate the secular equation 0 0 0 0 0 0 0 iA 0 0 0 0 0 - iA 0 0 - Δ E I = Δ E 0 0 0 0 Δ E 0 iA 0 0 Δ E 0 0 - iA 0 Δ E = 0 , whose solutions are Δ E = 0 , Δ E = 0 , Δ E = A, Δ E = - A .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern