HW #12 (221B), due Apr 29, 4pm
1. The classical energy for the Maxwell eld is H= dx 1 E2 + B2 . 8 (1)
(a) Show that the mode expansion, Ai (x) = Ai (x) = 2 c2 h L3 2 c2 h L3 1 p
p h ( i (p)a (p)eipx/ + h h (i i (p)a (p)eipx/ + i i (p) a (p)eipx/ ), ipx/
221B HW #2, due Jan 28 (Fri), 4pm
1. Solve the one-dimensional time-independent Schrdinger equation with o the potential V = (x) ( has the dimension of energy times length). (a) Find the solution of the form (k > 0) k (x) = eikx + Reikx T eikx (x < 0) (x
221B HW #3, due Feb 4 (Fri), 4pm
1. Consider a nucleus as a sphere with a uniform charge density. (a) Calculate the Rutherford scattering cross section of an electron by a nucleus together with the form factor using the rst Born approximation. (b) Compari
221B HW #4, due Feb 11 (Fri), 4pm
1. Consider the hard sphere scattering. Work out the phase shift exactly for arbitrary l, and plot sin2 l for ka = 0.1 and ka = 100. Explain the behavior using semi-classical arguments. Calculate the total cross section b
221B HW #5, due Feb 18 (Fri), 4pm
1. Study the scattering by a not-so-hard sphere V (r) = (r > a). Assume k > K.
2m K2 h 2
(r < a) and V (r) = 0
(a) Solve for the phase shifts exactly. Plot sin2 l for Ka = 10 and 3 and ka = 30 up to l = 50. Calculate the
HW #6 (221B), due Feb 25, 4pm
1. Consider an atom with three electrons, such as Li, Be+ , B+ . The Hamiltonian is H = H0 + H
3
(1) Ze ri
2
H0 =
i=1
p2 i 2m
(2) (3)
H = +
i<j
e2 . rij
The unperturbed Hamiltonian is the same as in the hydrogen-like atoms an
HW #7 (221B), due Mar 4, 4pm
1. The copper metal is a cubic lattice with the lattice constant of 3.61. Assume A that there is one conduction electron per lattice site and treat the electrons as free particles. Calculate the Fermi energy and show that it i
Take-Home Midterm Exam (221B), due March 18, 4pm
1. Consider two electrons in the np orbital. Answer the following question. (a) Show that the Coulomb repulsion between two electrons has the form e 1 np | |np2 = F 0 + 25 F 2 r12 0 12 F 5F
2 2
2 F 0 + 5F
HW #9 (221B), due Apr 8, 4pm
1. Lookup levels of 210 Pb, 210 Bi, and 210 Po, and explain the low-lying excitations using the shell model. 2. Lookup levels of the shell model.
17
O, and
17
F, and explain the low-lying excitations using
3. Consider 2s and 2
HW #8 (221B), due Apr 1, 4pm
1. Consider a one-dimensional problem of two heavy particles at x1 and x2 of mass M and one light particle at x3 of mass m attached by springs, with the light particle in the middle: x1 < x3 < x2 . The Hamiltonian of the syste
HW #10 (221B), due Apr 15, 4pm
1. Take the Schrdinger eld Lagrangian without the interaction, o L= h . dx i h 2m
2 2
(1)
1 h Rewrite the Lagrangian with the Fourier modes = p a(p) L3/2 eipx/ with the box normalization and p = h(nx , ny , nz )/L for nx , n
Take-Home Final Exam (221B), due May 13, 4pm
1. Consider the decay of a 3d state of hydrogen atom to the 2p level. (a) If the initial state has m = 2, show that the only possible nal state is m = 1. (b) Calculate the decay rate for this transition. (c) Co
HW #11 (221B), due Apr 22, 4pm
1. Suppose annihilation and creation operators satisfy the standard commutation relation [a, a ] = 1. (a) Show that the Bogliubov transformation b = a cosh + a sinh (1) preserves the commutation relation of creation and anni
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