HW 5 Solutions Problem 1 Taking the z axis to be the direction of the electric eld and using z = r cos , the only non-zero transition matrix element is between the ground state and the 2p state with ml = 0. Hence, H210,100 (t) = eE0 dr210 (r)r cos 100 (r)
Physics 137B (Quantum Mechanics)
Problem Set #2
Due by September 16, 2013
1
The harmonic oscillator as an example
Consider the Harmonic oscillator with Hamiltonian:
H=
12
p
2m
1
+ 2 k x2 ,
and set
k
m
.
Let |n denote the eigenstate with energy (n + 1 ) .
Physics 137B (Quantum Mechanics)
Problem Set #7, Due by October 28, 2013
The Variational Principle
1
V = 1 x6 potential
2
Find an upper bound on the ground state energy of the Hamiltonian
p2
+ 1 x6 ,
H=
2m 2
where > 0 is a constant. Use a trial wavefuncti
Physics 137B (Quantum Mechanics)
Problem Set #6, Due by October 21, 2013
Nondegenerate Perturbation Theory
1
3-state system
(Modied from problem 6.9 of Griths.)
Consider a quantum system with just 3 linearly independent states. The
Hamiltonian, in matrix
Physics 137B (Quantum Mechanics)
Problem Set #9, Due by November 12, 2013 (Tuesday)
1
Nuclear Magnetic Resonance and Rabi Oscillations
The 1944 Nobel prize in physics was awarded to Isidor Rabi for his resonance method for recording the magnetic propertie
Physics 137B (Quantum Mechanics)
Problem Set #8, Due by November 4, 2013
1
WKB approximation for energy levels
Use the WKB approximation in the form
x2
x1
1
p(x)dx = (n 2 ) ,
n = 1, 2, 3, . . .
to analyze the spectrum of the Hamiltonian H =
a constant:
p2
Physics 137B (Quantum Mechanics)
Problem Set #12, Due by December 13, 2013 (Friday)
1
Scattering
Consider the 3D spherically symmetric potential
V (r) =
V0 for r < R
0
for r > R
where V0 > 0 and R > 0 are constants. In this problem set we will calculate
t
Physics 137B (Quantum Mechanics)
Problem Set #10, Due by November 18, 2013 (Monday)
1
The Fermi Golden Rule
In class we studied the Fermi golden rule as it applies to
stimulated emission or absorption of radiation. But the Fermi
golden rule also applies t
Physics 137B (Quantum Mechanics)
Problem Set #11, Due by November 25, 2013 (Monday)
The Adiabatic Approximation
A particle of mass m is in an innite potential well
0 < x < a and is in the ground state |1 at time
t = 0. The right wall then starts moving sl
Physics 137B (Quantum Mechanics)
Problem Set #1
Due by September 9, 2013
1
Two-state systems
The simplest nontrivial quantum mechanical system has a Hilbert space of dimension two.
We pick |1 and |2 as an orthonormal basis, and this allows us to expand a
Physics 137B (Quantum Mechanics)
Problem Set #4
Due by September 30, 2013
1
Lorentz force from Ehrenfests law
The Hamiltonian of a (spin-0) particle of mass m and charge q in a static
magnetic eld B is
H = 21 (p q A)2 ,
m
where A is the vector potential (
Physics 137B (Quantum Mechanics)
Problem Set #5
Due by October 7, 2013
Identical particles
Two one-dimensional identical indistinguishable particles obey Schrdingers
o
equation with Hamiltonian:
p2
p2
1
1
H = 1 + 2 + 2 M (2 )2 ( x1 +x2 )2 + 2 2 (x1 x2 )2
HW 7 Solutions Problem 1:
h Neutrons interact with the external magnetic eld via their spin I = where are the Pauli 2 matrices. The Hamiltonian is similar to the electrons case up to a multiplicative constant due to a dierent g factor. Indeed, the Hamilto
HW 8 Solutions Problem 1: (i)We note that the dependence on for a p-wave is Pl () = cos(). Now, let Rl (r) = (r)/r, then the radial equation tells us that 1(1 + 1) d2 + k2 =0 2 dr r2 In our case, note that (r) = 1 + (1) is satised.
i kr
(1)
eikr . Substit
Homework 9 Solutions Problem 1 (i)First of all, prove that for any arbitrary density matrices i and j , Tr i j 1 with the upper limit being reached if and only if i = j is a pure density matrix. Each density matrix has its own spectral representation. (i)
Suggested Solutions to Midterm 1
Problem 1
(a)Expand the wavefunctions and energy eigenvalues in terms of the perturbation parameter λ:
(0)
(1)
(0)
(1)
H = H0 + λH , ψn = ψn + λψn + . . . , En = En + λEn + . . . Thus, keeping terms up to linear
order in λ
Physics 137B (Quantum Mechanics)
Problem Set #3
Due by September 23, 2013
1
Addition of angular momenta
In this exercise we will practice addition of angular momenta by calculating
the Clebsch-Gordan coecients for the addition of a spin s1 = 3 and a spin
Midterm Exam # 2
Physics 137B, Spring 2004
PLEASE MAKE SURE YOU WRITE YOUR NAME AND STUDENT ID ON
YOUR EXAM.
This exam contains 3 questions, each with multiple parts. You should answer all
the questions to the best of your ability. Please show your all wo
Physics 137B (Quantum Mechanics II) Midterm Exam / October 15, 2013 / 9:40am - 11:00am
Please solve all three problems below, and please explain your answers.
Problem 1 [30pts]: Spin-orbit interaction
2
1
e
The spin-orbit interaction Hso = 8 0 m2 c2 r3 S
Physics 137B (Professor Shapiro) Spring 2010
GSI: Tom Grin
Homework 9 Solutions
1. From equation 12.94 of the text, the energy levels of the hydrogen atom,
taking into account ne structure and a weak-eld Zeeman splitting, are
|n, l, j, mj > with energy:
E
Physics 137B (Professor Shapiro) Spring 2010
GSI: Tom Grin
Homework 7 Solutions
1. We have that:
Hn0 (t) = qE(t) < n|x|0 >
h
n1
2m
= qE(t)
Thus only a transition to the rst excited state is allowed. Putting this into
equation 9.17 of the text, we obtain:
Physics 137B (Professor Shapiro) Spring 2010
GSI: Tom Grin
Homework 6 Solutions
2
L
1. For t < 0, eigenstates are n (x) =
sin( nx ). For t > 0 the well has width
L
2
2L and the eigenstates are n (x) = 2L sin( nx ). For t < 0 the system is in
2L
the state