Ph125c lecture notes, 4/5/01
Introducing the spectroscopic notation nL j , we have thus found the fine-structure energy
corrections
5 mc 2 4 ,
E 2s 1/2
128
5 mc 2 4 ,
E 2p 1/2
128
1 mc 2 4 .
E 2p 3/2
128
[Cohen-Tannoudji et al Ch. XII Fig. 2]
The hyperfin
Ph125c - Problem Set 6
Due: Wednesday, May 25th 2016
Problem 1
Consider a Dirac spinor . Derive the continuity equation,
+ ~ = 0,
t
where =
and the current density is ~ =
c
~ ,
with
~=
0 ~
~ 0
.
Problem 2
~
Consider the kinetic momentum operator ~ = P~
Physics 125c Problem Set 2 Solutions
Problem 1
The energy of the particle is,
E=
p2
kT ,
2m
which gives us,
p
2mkT .
Thus, the de Broglie wavelength is,
=
h
h
.
p
2mkT
The thermal wavelength is given by,
r
1
h
~=
.
mkT
2 mkT
The two wavelengths are the
Ph125c - Problem Set 4
April 21, 2016
Problem 1
Consider the P
oschl-Teller theory, defined for a particle of mass m with a potential,
V (x) =
~2 a 2
sech2 (ax)
m
for a constant a.
(a) For a ground state ansatz,
state energy, Eground .
ground (x)
= A sech
Physics 125c Midterm Due May 6, 2016
Examination Rules
You are allowed a total of 4 continuous hours to take this exam. The exam is due by 5pm on
Friday, May 6, in the usual problem set box. You may take this exam during any continuous 4-hour
time slot be
Physics 125c Problem Set 5
due May 18, 2016
Problem 1 (10 points)
Suppose we are given some A and that do not obey the Coulomb gauge conditions. Let us see
how they can be transformed to the Coulomb gauge.
(a) Show that if we choose,
Z
t
(r, t) = c
(r, t0
Ph125c - Problem Set 3
Problem 1
Show that the wave function of the coherent state is,
z (x) = hx|zi =
m 1/4
~
z 2 m 2
exp
x +
2
2~
r
!
2m
zx .
~
(1)
Start by using a |zi = z |zi in the coordinate representation and fix the normalization by demanding
0
Problem Set 3 Solutions
Ph125c - Spring 2015
Problem 1
In the coordinate representation, a |zi = z |zi becomes
r
m
2~
i
p z (x) = zz (x)
x+
m
r
z (x) =
x
!
2m
m
z
x z (x) ,
~
~
the solution of which is
r
z (x) = A(z) exp
2m
m 2
zx
x
~
2~
We fix the norm
Ph125c - Problem Set 1 Solutions
April 13, 2015
Problem 1
For potential V (x) = f x, the classical solution is given by:
xcl ( ) =
f 2
+ v0 + x0 .
2m
For initial conditions xcl (0) = x0 and xcl (t) = x, we have:
x0 = x0 ,
v0 =
f
x x0
t+
.
2m
t
The class
Problem Set 6 Solutions
Ph125c - Spring 2015
Problem 1
The Dirac equation for is
i~
~ + mc2 ,
= ic~
~
t
and the equation for the conjugate spinor is
i~
~ )
= ic~ (
~ + mc2 .
t
Thus, we find that
2
mc2
~ )
~ i mc
=
+
= c(
~ + i
c~
t
t
t
~
~
Ph125c lecture notes, 4/12/01
Coupling to a continuum; Fermi s Golden Rule
So far in our discussion we have implicitly assumed that | k and | s lie in a discrete (bound)
part of the eigenspectrum of H 0 , as opposed to a continuum. Transitions from bound
Ph125c lecture notes, 4/19/01
Many-body systems; Bosons and Fermions
The topic of many-body systems and particle statistics properly belongs to quantum field
theory, but it is traditional to attempt to treat these subjects within the paradigm of standard
Homework projects
The assignment is to prepare a set of lectures notes, comparable in length and level of technical detail to my notes from a good class, on an advanced topic/application of the quantum mechanics you have learned this year. Use the class n
Ph125c lecture notes, 5/24/01
More on generalized measurement
Let's revisit a problem we considered at the very beginning of this course, on the optimal measurement to distinguish between two non-orthogonal states. Say we have a quantum system A whose sta
Ph125c lecture notes, 5/31/01
Selective evolution and the Stochastic Schrdinger Equation
Consider the following general form for a pair of operations form a complete measurement:
0,
0
1
1. Here we assume that H is a Hermitian operator, and that the scalar
Ph125c lecture notes, 4/3/01
About the third term
Syllabus
The Hydrogen atom and other alkali atoms Bound-state perturbation theory; Fine, Hyperfine, and Zeeman structure Time-dependent perturbation theory Scattering, partial waves Identical particles Pho
Ph125c lecture notes, 4/10/01
Time-dependent perturbation theory
Say we have a time-dependent Hamiltonian
Ht
H0
Wt.
There will not generally be stationary states for the corresponding Schrdinger Equation,
because of the explicit time-dependence. To solve
Ph125c lecture notes, 4/17/01
This class was a review of partial wave scattering and its relation to the problems we
studied on sectionally constant potentials in 1D (see Ph125b lecture notes from 1/23/01).
We also looked at the research paper Direct obse
Ph125c lecture notes, 4/24/01
Fermi degeneracy and the electron gas
Today we consider the effect of quantum statistics on the free electron gas, free in the
sense that we will neglect electron-electron interactions and assume only a box-like
confining pot
Ph125c lecture notes, 5/8/01
Photons and quantization of the electromagnetic field
A good general reference for these topics is C. Cohen-Tannoudji, J. Dupont-Roc, and G.
Grynberg, Photons and AtomsIntroduction to Quantum Electrodynamics (John Wiley &
Sons
Ph125c lecture notes, 5/22/01
Generalized measurement theory
Early on in this course (start of first term) we discussed the basic formalism of quantum measurement, in terms of projection operators. Today we'll develop a broader picture, introducing first