Phys 623, Introduction to Quantum Mechanics II, Spring 1999, Dr. Yakovenko

Final Examination Saturday, May 22, 1999, 8–10 a.m., room 1410

1. [10 points] The shell model of nuclear structure assumes that protons and neutrons move in an effective spherically-symmetrical potential well produced by nuclear forces. The energy levels are labeled by the orbital angular momentum l and the radial quantum number N. The lowest energy levels are 1s and 1p (see p. 317 of Schwabl). These levels are completely filled by 2+6=8 protons and 2+6=8 neutrons in the oxygen nucleus _{8}^{16}O_{8} (see the nomenclature on p. 318 of Schwabl). Obviously, because the energy shells are completely filled, the oxygen nucleus has zero magnetic moment.

Calculate the magnetic moment (in nuclear magnetons μ_{N}) of the _{7}^{15}N_{8} nitrogen nucleus. In this nucleus, one proton in the ^{2}P_{1/2} state (see the notation on p. 217 of Schwabl) is missing from the 1p shell compared to oxygen. The magnetic moment of a free proton is μ_{p} = 2.8μ_{N}.

2. An electron of mass m is in the ground state of a spherically-symmetrical harmonic potential of frequency Ω. The electron is subject to a weak spatially-uniform timedependent electric field polarized along the z axis and being gradually turned on over a time τ:

E(t) = \{ \begin{matrix}2e_{z}E_{0} e^{(t−t_{0})/τ} cos(ωt), & t < t_{0} \\ 2e_{z}E_{0} cos(ωt), & t > t_{0} \end{matrix}  (1)

(a) [3 points] Calculate the matrix element of the perturbation between the ground state and an excited electron state.

(b) [4 points] In the first order of perturbation theory, calculate the probability P (t) of electron transition to an excited state at t < t_{0}. In the answer for P (t), there may be some terms that oscillate in time with the frequency ω. Average those terms over time, assuming that 2π/ω ≪ Δt ≪ τ, where Δt is the time of observation. Sketch the answer for the averaged probability P̄ at t = t_{0} as a function of ω, assuming that 1/τ ≪ ω, Ω.

(c) [4 points] Using the result of Problem 2b, calculate the transition rate Γ = dP̄(t)/dt at t = t_{0}. Show how the Fermi golden rule in recovered in the limit τ → ∞.

3. Qualifier, January 1999, II-3

A particle of mass m scatters from a spherically symmetric potential

V (r) = \frac{A}{r^{2}} ,  (2)

where A is a constant.

(a) [6 points] Find the scattering amplitude f(θ) in the Born approximation.

(b) [6 points] Find the exact partial wave phase shifts δ_{l}.

Hint: V (r) looks a lot like the centrifugal potential.

(c) [6 points] Show that the Born approximation is valid at any energy provided the condition mA/h̄^{2} ≪ 1 holds. Suggestion: You may want to replace V (r) by the finite range potential V (r)e^{−r/a} and take the limit a → ∞.

(d) [6 points] Show that your results for parts 3a and 3b give the same value for

f(θ) in the limit mA/h̄^{2} ≪ 1.

Possibly useful information:

The asymptotic forms of the spherical Bessel and Neumann functions j_{ν}(x) and n_{ν}(x) for any fixed complex ν and |x| → ∞ are

j_{ν}(x) ∼ \frac{1}{x} sin(x − νπ/2), n_{l}(x) ∼ −\frac{1}{x} cos(x − νπ/2).

Possibly useful integral and sum:

∫_{0}^{∞} \frac{sin x}{x} dx = \frac{π}{2} ,  ∑_{n=0}^{∞} P_{n}(cos θ) = \frac{1}{2 sin(θ/2)}.