HW 12 Ans Key
Song Zhang
Dec 14th, 2016
1. The first order Born formula tells us
m
f () =
d3 rV (r)eiqr
2~2
mV0
= 2 i sin(qz )
~
(1)
(2)
where q = k k. qz = k(1 cos ) = 2k sin2 ( 2 ).
Now for the validity of Born approximation, we must require the pertur
HW 10 Ans Key
Song Zhang
Dec 5th, 2016
1. The transition rate with incoherent light source is given by equation
R=
2
|d|
()
30 ~2
(1)
where () is the energy density of the field. For thermal radiation
field, then density is
() =
~
3
~
2 c3 e kB T 1
(2)
2
HW 8 Ans Key
Song Zhang
Nov 25, 2016
1. Using the quantization formula from Griith
$2 1
/ p(a:)da: = (n )7rh with n = 1,2,3. (1)
1171
Since the potential is symmetric, the turning points is also symmetric.
It is given by the equation.E = V(x2) or $2 = a+
HW 11 Ans Key
Song Zhang
Dec 8th, 2016
1.
a) Selection rule tells us l = 1 and m = 0, 1. So the only
possible transition is |311 |100.
b) 100%
2
|d|
c) Spontaneous transition rate is given by A = 3
3 , where is the
0 ~c
radiated photon frequency between
HW 9 Ans Key
Song Zhang
Nov 25, 2016
1.
a) The first order transition probability is given by
1
P01 = |
ei10 t H10
dt|2
~ 0
(1)
The
perturbation for this question is H = eEx.
Then H10 =
2E2
~
k
and P01 = 2e
sin2 (
), where = m
.
eE 2m
m~ 3
2
b) The fi
QUANTUM MECHANICS
Charles G Wohl
I can play every note that Jimi Hendrix ever played.
And I can play each note exactly the way he played it.
But what I cant imagine is how he thought of it in the first place.
Steve Vai, as quoted in Guitar Zero, by Gary M
12. ENTANGLEMENT
1. Quantum Cryptography
2. The EPR Argument
3. Bells Inequality
Problem
A short chapter, based entirely on the |S, M = |0, 0 state of two spin-1/2 particles:
)
1 (
|0, 0 = |+ |+ .
2
The kets on the right side give the |m1 , m2 states of t
6. THE SIMPLE HARMONIC OSCILLATOR
1. The Classical Oscillator
2. The Quantum Oscillator: Series Solution
3. The Operator Solution
4. States as Vectors, Operators as Matrices
Problems
Essentially, all models are wrong, but some are useful.
George E.P. Box
15. TIME-DEPENDENT PERTURBATION THEORY
1. Sudden Changes
2. Time-Dependent Perturbation Theory
3. Magnetic Resonance, Again
4. Oscillator in a Time-Dependent Field
5. Hydrogen in an Electromagnetic Wave
6. Averaging over Polarizations and Frequencies
7. T
4. SCATTERING IN ONE DIMENSION
1. Particle Densities and Currents
2. Scattering by a Step
3. Scattering by a Rectangular Barrier
4. An Approximation for Weak Tunneling
5. Multiple Barriers (not written)
Problems
Chapter 2 was almost entirely about bound s
3. SIMPLE APPROXIMATIONS
1. Dimensions and Scaling
2. Fitting Wavelengths in a Well: the Semiclassical Method
3. Guessing the Ground State: the Variational Method
Problems
In this chapter, we develop three ways to get partial or approximate information ab
11. TWO ANGULAR MOMENTA
1. Hyperfine Structure of the Hydrogen Ground State
2. The 21-cm Line and Astronomy
3. Total Spin of Two Spin-1/2 Particles
4. Coupling Any Two Angular Momenta
5. Clebsch-Gordan Coecients
6. Particle Multiplets and Isospin
Problems
HW 2 Ans Key
Song Zhang
October 08, 2016
1.
a) Lets solve for the dynamics of this system. The state vector will
Ht
evolve as |(t) = ei ~ |(0), where the dipole field interaction
We have magnetic moment
Hamiltonian H is given by H = B.
and field is B
=
HW 2 Ans Key
Song Zhang
October 08, 2016
1.
~
a) Introduce a few notations. Let n = n2ma
be the energy levels
2
and |n be the corresponding eigenstate for single particle. Then
ground state for distinguishable particles is
2 2 2
|1 = |111
with energy E1
HW 1 Ans Key
Song Zhang
October 08, 2016
1. Discussed in tutorial session, skip.
2.
)
(
cos sin
a) The spin matrix along the direction n
reads Sn =
.
sin cos
The eigenvalues are clearly ~2 . The corresponding eigenvectors
are |+ = cos 2 | + sin 2 |da a
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
Midterm Exam # 1
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: Midterm 1
September 27, 2011
Problem 1: Variational Method (5 points)
Use a Gaussian trial function to obtain the lowest upper bound you can on the ground state
energy of the linear potential V (x) = |x|.
Hint: The following integrals may be
Physics 137B: Midterm 1 Solutions
September 27, 2011
Problem 1: Variational Method (5 points)
Use a Gaussian trial function to obtain the lowest upper bound you can on the ground state
energy of the linear potential V (x) = |x|.
Hint: The following integr
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
gamma MT #4»
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m SH 1- (SYMNE)
m 1%. Mia/93?} l~-<.[=/,3'-~
By] QWQJ:/)C"l) 6L, mWe/Wéjl/ (a) ror the symmetric case, we can have
Homework 2 Problem 5
Part a
The path element is given by:
d5 = 1; dz? + dy'2 + dz2
d:r+dy
@ 3h )3
3: 3y
= d+dy2+(
3h 2 3h 2 3131
= [1 + (31:) Jirg + [1+ (a) :|dy2 +23Igdldy
Therefcx'e:
3:
f(y1y1l=
Part b
For May = A + BI + By, then:
fiy1y1z= 1! [1+
Part a
From conservation of energy:
1 Part (1
mu2 _ I = 0 _ 1
2 T T Isolating a
21: 1 1
= _ 1 1 1 1
1! 171(1! To) 02 (_) 272(1) 02 (_) $12
7' To T To
1 _ A
Part b r2 = L
r2 02 - oz (1- L
The path element in polar coordinates is given by: T '1
d3 = cfw_if-