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/
HW6.nb
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HW #6
1. Three-electron Atoms
(a)
We set up the Slater determinant for three electrons, i 1 s \1 1 s \2 1 s \3 y j z j z j z 1 j z 1 s2 2 s\ = detj 1 s \1 1 s \2 1 s \3 z ! j z j z 3! j z j z 2 s\2 2 s\3 cfw_ k 2 s\1 Here, the subscripts refer to
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
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
HW5.nb
1
HW #5
1. Not-so-hard sphere
(a)
! We solve the problem exactly. The wave function is Rl HrL = jl Hk rL cos dl + nl Hk rL sin dl for r > a, and Rl HrL = jl I k2 - K 2 rM ! for r < a at high energies k > K = 2 m V , . Requiring the logarithmic deri
HW4.nb
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HW #4
1. Hard Sphere Scattering
For the hard sphere scattering, the requirement on the radial wave functionn is Rl HrL = jl Hk rL cos dl + nl Hk rL sin dl for r > a, jl Hk aL tan2 q and Rl HaL = 0. Therefore, tan dl = - . Using the trigonometric
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
HW3.nb
1
HW #3
1. nuclear size
(a)
As shown in the lecture notes Scattering Theory II, the form factor is sin q a-q a FHqL = 3 aL3cos q a . Hq The zeros of this function can be found numerically by
Sin@xD - x Cos@xD PlotA3 , 8x, 0, 20<E x3
0.15 0.1 0.05
5
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
HW2.nb
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HW #2
1. 1D scattering
(a)
2mm The delta function potential requires a discontinuity of the wave function, y' H+0L - y' H-0L = yH0L, while the wavefunc2 tion itself is continuous.
2mm SolveA91 + R T, I k T - I k H1 - RL T=, 8R, T<E 2 mm k 2 99R -
HW9.nb
1
HW #9
1. A=210
These nuclides add two nucleons to the double-magic 208 Pb. For 210 Pb, we add two neutrons into the 2 g92 orbital. Taking the proper anti-symmetry of two neutrons into account, the possible spin-parities are 0+ , 2+ , 4+ , 6+ , 8+
HW7.nb
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HW #7
1. Fermi energy
This is a problem meant to remind ourselves Fermi-degenerate gas which is assumed in the Thomas-Fermi model of atoms, not a realistic condensed-matter problem which is beyond the scope of this course. In any case, here we go
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
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
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
HW12.nb
1
HW #12
1.
(a)
Using the given mode expansion of the vector potential, we check the Coulomb gauge condition. Acting the divergence on the vector potential simply pulls out i p for the term with annihilation operators or -i p for the term with cre
HW10.nb
1
HW #10
1. Schrdinger field in momentum space
We start with the Lagrangian (sorry that y* was missing in the second term in the problem set)
- L = d x Ay* i y - y* m yE. 2 We subsitute
2 2
y = aIpM 1 ei px . L32
p
The first term is
1 -i p x *
HW11.nb
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HW #11
1. Bogoliubov Transformation
(a)
This is a straight-forward algebra @b, b D = @a cosh h + a sinh h, a c osh h + a sinh hD = @a, a D cosh2 h + @a , aD sinh2 h = cosh2 h - sinh2 h =1
(b)
Using b, b defined in (a), we can solve for a, a as a
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
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 #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
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
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