Section 3.13: Theorems for Hermitian Operators (under construction) This section discussions some theorems for operators (refer to T. D. Lees Book). The common theorems show when two operators are equal and also discusses alternate methods for determining
Zachary Fisher
February 27, 2017
Physics 137B Midterm Review Solutions
1.
Short questions
(a) For what kind of problems is first-order perturbation theory a good approximation?
Solution. First-order perturbation theory works well if a quantum system under
Section Notes 8. Partial Wave Scattering
April 21, 2017
1.
Useful formul
Y`0 (, ) = Y`0 () is real and independent of .
Z
Y`1 0 ()Y`2 0 () d = `1 ,`2
r
2` + 1
Y`0 (0) =
4
r
2` + 1
Y`0 () =
P` (cos )
4
Here, Y`m are the spherical harmonics, and P` is the `
Section Notes 4. Selection Rules
February 21, 2017
Selection rules: we begin by considering what seems to be a very dry problem, which is to evaluate things
like
hn0 l0 m0 |z|nlmi = hn0 l0 m0 |r cos |nlmi = hn0 l0 |r|nli hl0 m0 |cos |lmi =?
(0.1)
for hydr
3. Degenerate Perturbation Theory
February 15, 2017
1.
A simple 3x3 example [Griffiths Problem 6.9]
Consider a quantum system with just three linearly independent states. Suppose the Hamiltonian, in
matrix form, is
(1 ) 0
= V0 0
1
H
0
0
2
(1.1)
Here, V0
Section Notes 2. Perturbation Theory
February 8, 2017
1.
Useful Formulas
For a harmonic oscillator:
r
~
(
a + a
+ )
2m
r
~m +
p = i
(
a a
)
2
r
p
m
a
=
x
i
2~
m
+
n
n+1
a
| i = n + 1
a
| n i = n n1
x
=
En = ~(n +
1
2)
General non-degenerate perturb
Angular momentum
February 8, 2017
Review of angular momentum
S2 |s, mi = ~2 s(s + 1) |s, mi
Sz |s, mi = ~m |s, mi
(0.1)
S+ = Sx + iSy
S = Sx iSy
~1 S
~2 = 1 (S + S + S S + ) + S1z S2z
S
1 2
2p 1 2
S |s, mi = ~ s(s + 1) m(m 1) |s, m 1i
[Sz , S ] = ~S
(0.3)
Section Notes 7. Scattering
April 4, 2017
Review of definitions:
1. Solid angle: e.g. what fraction of the night sky is covered in stars? (Distance to star is irrelevant
here.) Multiply fraction by 4 to get solid angle.
2. Scattering axis: axis of dead-pe
March 14, 2017
Physics 137B: Time-Dependent Perturbation Theory
Useful information. Time-dependent perturbation theory is used when a perturbation depends on
0 are called |ni with energies En0 , an arbitrary state evolves as
time. If the eigenstates of H
Section Notes 5. Estimation Methods and the Variational
Principle
March 9, 2017
Consider the potential V (x) = |x|, where > 0 is a constant. We need to solve for (and E) from
Schrodinger equation
~2 d2
+ |x| = E
2m dx2
(0.1)
This equation can be solved e
9. SPIN-1/2 PARTICLES
1. Spinors. Eigenvalues and Eigenstates 2. The Polarization Vector 3. Magnetic Moments and Magnetic Fields 4. Time Dependence. Precessing the Polarization 5. Magnetic Resonance. Flipping the Polarization 6. Stern-Gerlach Experiments
4. MATHEMATICAL FORMALISM
1. Vector Spaces. Dirac Notation 2. States as Vectors 3. Operators 4. Successive Operations. Commutators 5. Operators as Matrices 6. Expectation Values 7. More Theorems 8. Rules and Interpretations Problems
This chapter needs a l
1. PARTICLES AND WAVES. THE ATOMIC MODEL
1. Particles and Waves: Neutrons and Light 2. The Bohr Hydrogen Atom: Quantized Energies and Spectra Problems
It will be assumed that the reader has some background in modern physics. This chapter is not meant to b
Formalism of Quantum Mechanics
Dirac Notation We can use a shorthand notation for the normalization integral
I = " ! (r, t ) dr = " ! * (r, t )! (r, t )dr = ! !
2
The state ! is called a ket. The complex conjugate of the ket ! is called a bra More general
Hermitian Operators
Definition: an operator is said to be Hermitian if it satisfies: A=A
Alternatively called self adjoint In QM we will see that all observable properties must be represented by Hermitian operators
Eigenvectors of a Hermitian operator
Harmonic Oscillator: From a and a to the position representation
Michele Kotiuga Physics 137A Spring 2010
In lecture the harmonic oscillator potential was treated using creation and annihilation operators: a and a . In Section 4.7 of Bransden and Joachain
Physics 137B (Quantum Mechanics), Spring 2011 Please read this page carefully! Pages 25 are the contents of a text, most of which is written. Pages 6 and 7 oer some advice. Lecturer Charles Wohl, [email protected] 486-4730, oce hours in 251 Le Conte (the Rea
Zachary Fisher
May 8, 2017
Physics 137B Final Exam Review Solutions
Bill Watterson
1.
Conceptual questions
(a) Explain why the variational method is useful for studying the ground state energy of Helium. What
tricks can be employed to get the best possibl