Quantum Mechanics II
Problem Set 1
Due: 13 January 2016
Reading: Shankar, Chapter 8 and SEctions 12.1-12.3; Lecture Notes, Chapter 10 and begin
reading Chapter 11.
1. Path Integral for the Harmonic Oscillator. The Lagrangian for a forced harmonic oscillat
Quantum Mechanics II
Problem Set 1
Due: 14 January 2015
Reading: Shankar, Chapter 12 and Lecture Notes, Chapter 11.
1. We have chosen a basis of the Lie algebra of SO(3) as
0 0 1
0 0 0
J2 = 0 0 0 ,
J1 = 0 0 1 ,
1 0 0
0 1 0
a) Check the commutator algebra
Classical Mechanics
Action and Hamiltonian
Z
I =
t2
dtL(qk (t), qk (t), t),
H
t1
X
qi
i
L
L
qi
Particle of charge Q in EM field
L=
m 2
r + Q(r A ),
2
B = A,
E = A
Hamiltons equations
pk =
H
,
qk
qk =
H
pk
Poisson Bracket
X f g
f g
cfw_f, g
qk pk pk qk
Quantum Mechanics
Charles B. Thorn1
Institute for Fundamental Theory
Department of Physics, University of Florida, Gainesville FL 32611
Abstract
1
E-mail address: thorn@phys.ufl.edu
Contents
1 Introduction
5
2 General Formulation of Quantum Mechanics
2.1
Quantum Mechanics II
Problem Set 2
Due: 20 January 2016
Reading: continue studying Shankar, Chapter 12 and Lecture Notes, Chapter 11.
5. Prove that any rotation in three dimensions leaves an axis invariant. You can do this
by proving that any orthogonal m
Quantum Mechanics II
Problem Set 6
Due: 17 February 2016
Reading: Shankar, Chapter 15 and Lecture Notes, Chapter 13
26. a) S, Exercise 14.5.II (top of page 399. This problem should have had the number
14.5.2). b) S, Exercise 14.5.3
27. S, Exercise 15.1.2
Quantum Mechanics II
Problem Set 5
Due: 10 February 2016
Reading: Shankar, Chapter 14 and Lecture Notes, Chapter 12
21. a) S, Exercise 13.4.1; b) S, Exercise 13.4.2; c) S, Exercise 13.4.3,
22. a) S, Exercise 14.3.7 (Use projectors (I x )/2 onto the eigens
Quantum Mechanics II
Problem Set 8
Due: 16 March 2016
Reading: Shankar, Chapter 17 and Section 16.1 and Lecture Notes, Chapters 14,15
36. In this problem we calculate the effect on the n = 3 Coulomb energy levels of hydrogen
due to a uniform electric fiel
Quantum Mechanics II
Problem Set 9
Due: 23 March 2016
Reading: Shankar, Sections 16.1 and 18.1-18.3 and Lecture Notes, Chapters 15, 16
40. a) S, Exercise 16.1.2; b) S, Exercise 16.1.3
41. S, Exercise 16.1.5.
42. Perform a variational calculation for the g
Quantum Mechanics II
Problem Set 7
Due: 9 March 2016
Reading: Shankar, Chapter 17 and Lecture Notes, Chapter 14
31. From previous studies of the forced oscillator, we know that the Hamiltonian
H = ~(a a + 1/2) + f a + f a
can be exactly diagonalized. For
Quantum Mechanics II
Solution Set 11
Due: 6 April 2016
Reading: Shankar, Sections 19.1-19.4 and Lecture Notes, Chapters 16 and 17
51. S, Exercise 18.2.3.
52. S, Exercise 18.2.4. This problem uses the sudden approximation, described on p. 477 of
Shankar.
5
Quantum Mechanics II
Problem Set 10
Due: 30 March 2016
Reading: Shankar, Finish Chapter 18 and Lecture Notes, Chapter 16
45. S, Exercise 18.2.6
46. In class we found that to first order in a harmonically varying potential VS = V0 eit +
V0 eit the transiti
Quantum Mechanics II
Problem Set 12
Due: 13 April 2016
Reading: Shankar, Chapter 19 and Lecture Notes, Chapter 17
57. Do the followiing Born approximation problems: a) S, Exercise 19.3.1; b) S, Exercise
19.3.2; c) S, Exercise 19.3.3
58. Consider N static,
Quantum Mechanics II
Problem Set 13
Due: 20 April 2016
Reading: Shankar, Continue Chapter 19 and Lecture Notes, Chapter 17
62. S, Exercise 19.5.1
63. a) S, Exercise 19.5.3; b) S, Exercise 19.5.5.
64. Consider the scattering of a particle in the delta shel
Quantum Mechanics II
Problem Set 3
Due: 27 January 2016
Reading: Shankar, Chapters 12 and Lecture Notes, Chapter 11.
10. a) S, Exercise 12.5.2; b) S, Exercise 12.5.3.
11. a) S, Exercise 12.5.12; b) S, Exercise 12.5.13.
12. Consider a single particle wave
Quantum Mechanics II
Problem Set 4
Due: 3 February 2016
Reading: Shankar, Chapters 13 and Lecture Notes, Section 11.7-11.10
14. (Note: This problem is moved from Problem Set 3.
Set up the transcendental equations that determine bound state energy levels f
Quantum Mechanics II
Solution Set 9
Due: 25 March 2015
Reading: Shankar, Sections 18.1-18.3 and Lecture Notes, Chapters 16
39. In our discussion of the helium atom, we assumed that the nucleus was a xed center of
force. In this problem we analyze the eect
Quantum Mechanics II
Solution Set 10
Due: 1 April 2015
Reading: Shankar, Finish Chapter 18 and Lecture Notes, Chapter 16
43. S, Exercise 18.4.3.
Solution:
(1) The gauge transformation is A = A and = + /c. With the proposed ,
= c(r, t), so it is immediate
Quantum Mechanics II
Solution Set 4
Due: 4 February 2015
Reading: Shankar, Chapters 13 and Lecture Notes, Section 11.9
16. a) Estimate the size and binding energy of the ground state of positronium, an atom
in which the nucleus is replaced by a positron,
PHY 6646 - Quantum Mechanics II - Spring 2012
Homework set # 4, due February 8
1. Show that
(1)
x
0
1
=2
2
0
0
1
= i 2
2
0
(1) =
z
1
0
0
0
2
0
i 2
0
i2
(1)
y
2
0
2
0
0
0
0
i 2
0
0
0
1
(0.1)
and
(1)
x
0
=0
0
(1)
y
0
= 0
i
0
0
0
0
= +i
0
i
0
0
(1)
z
0
0
+i
Diracs Postulates of Quantum Mechanics
I. Each pure state of a dynamical system is associated with a vector in a complex vector
space called State Space or Ket Space, K. The symbol for a ket vector corresponding
to a state A is |A .
II. Superposition Prin
Syllabus for Quantum Mechanics I & II, PHY 66456646
First Semester, PHY6645
1. Introduction: 2-slit quantum interferenc
2. Mathematical Review and Diracs Postulates of Quantum Mechanics
(a) Vector Spaces, QM Postulates I-III.
(b) Dual Vector space and Inn
Quantum Mechanics II
Problem Set 5
Due: 11 February 2015
Reading: Shankar, Chapter 14 and Lecture Notes, Chapter 12
21. S, Exercise 14.3.8.
Solution:
(1) A 2 2 matrix that commutes with z commutes with all diagonal matrices and hence
must be diagonal. It
Quantum Mechanics II
Solution Set 6
Due: 18 February 2015
Reading: Shankar, Chapter 15 and Lecture Notes, Chapter 13
25. a) S, Exercise 14.5.II (top of page 399. This problem should have had the number
14.5.2). b) S, Exercise 14.5.4
Solution:
a) (1) The s