PHYS851 Quantum Mechanics I, Fall 2008
HOMEWORK ASSIGNMENT 7: 1D Scattering: 1. [10pts]Step-down: An incident wave with wave-vector k1 approaching from the left encounters a step-down potential, V (x) = V0 u(x) where u(x) is the unit step function. Calcul
PHYS851 Quantum Mechanics I, Fall 2008
HOMEWORK ASSIGNMENT 2: Postulates of Quantum Mechanics
1. [10 pts] Assume that A|n = an |n but that n |n = 1. Prove that |an = c|n is also an eigenstate of A. What is its eigenvalue? What should c be so that an |an =
PHYS851 Quantum Mechanics I, Fall 2008
HOMEWORK ASSIGNMENT 7: 1D Scattering: 1. [10pts]Step-down: An incident wave with wave-vector k1 approaching from the left encounters a step-down potential, V (x) = V0 u(x) where u(x) is the unit step function. Calcul
PHYS851 Quantum Mechanics I, Fall 2007
HOMEWORK ASSIGNMENT 2: Postulates of Quantum Mechanics
1. [10 pts] Assume that A|n = an |n but that n |n = 1. Prove that |an = c|n is also an eigenstate of A. What is its eigenvalue? What should c be so that an |an =
HOMEWORK ASSIGNMENT 1
PHYS851 Quantum Mechanics I, Fall 2008
1. What is the relationship between | and | ? What is the relationship between the matrix elements of M and the matrix elements of M ? Assume that H = H what is m|H |n in terms of m|H |n ? 2. Pr
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 1
1. [25 points] This problem shows you how to derive the matrix representations of spin operators from rst principles. a.) For spin 1/2, use the eigenvalue equation Sz |ms = ms |ms to nd the c
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 2: Solutions
1. Consider the most general normalized spin-1/2 state: | = c+ |+ + c | , where Sz | = 2 | . a.) Compute Sx , Sy and Sz . b.) Compute the variances Sx , Sy , and Sz . c.) Prove tha
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 3:
1. Cohen-Tannoudji problem 10.1, page 1086 2. Cohen-Tannoudji problem 10.5, page 1087 3. The exact normalized eigenvalues and eigenstates of the hamiltonian H = Sz + Sx are 1 2 + 2 = 2 2 + 2
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 3: Solutions
1. Cohen-Tannoudji problem 10.1, page 1086 Answer: a.) For the 1s ground state, we have = 0, s = 1/2, and i = 1. Thus the possible values of j run from jmin = | s| = 1/2 to jmax =
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 4:
1. [20 points] Consider the shifted harmonic oscillator: H= P2 1 + m 2 X 2 + aX. 2m 2
Use perturbation theory to compute the eigenvalues to second order in a and the eigenstates to rst-order
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 4:
1. [20 points] Consider the shifted harmonic oscillator: H= P2 1 + m 2 X 2 + aX. 2m 2
Use perturbation theory to compute the eigenvalues to second order in a and the eigenstates to rstorder.
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 5:
1. [20 pts] The goal of this problem is to compute the Stark eect to rst-order for the n = 3 level of the hydrogen atoms. The stark shift is governed by the potential: VE = eE0 Z, so that yo
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 5: Solutions
1. [20 pts] The goal of this problem is to compute the Stark eect to rst-order for the n = 3 level of the hydrogen atoms. The stark shift is governed by the potential: VE = eE0 Z,
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 2
1. Consider the most general normalized spin-1/2 state: | = c+ |+ + c | , where Sz | = 2 | . a.) Compute Sx , Sy and Sz . b.) Compute the variances Sx , Sy , and Sz . c.) Prove that Sx = 2 |c
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 1: Solutions
1. [25 points] This problem shows you how to derive the matrix representations of spin operators from rst principles. a.) For spin 1/2, use the eigenvalue equation Sz |ms = ms |ms
Phys 852, Quantum mechanics II, Spring 2009
Time-Independent Perturbation Theory
Prof. Michael G. Moore, Michigan State University
Atomic Physics Applications
1
Introduction
For many reasons it is important to understand the basic level-structure of atomi
Discrete Translation Symmetry
Suppose the potential is periodic
Let a be the period:
a V(x)
Lecture 33: Symmetry IV: Lattice Symmetry
Formally this means:
V ( x + a) = V ( x)
In terms of the shift operator, this means:
PHY851 Quantum Mechanics I Fall,
PHYS851 Quantum Mechanics I, Fall 2008
HOMEWORK ASSIGNMENT 3: Fundamentals of Quantum Mechanics
1. Cohen-Tannoudju: pp 205-209, problems 8,9,10 pp 341-350, problems 1,3 2. Consider the system with three physical states cfw_|1 is: 1 H = 2i 1 , |2 , |3 . In
HOMEWORK ASSIGNMENT 1
PHYS851 Quantum Mechanics I, Fall 2008 1. [10 pts]What is the relationship between | and | ? What is the relationship between the matrix elements of M and the matrix elements of M . Assume that H = H what is m|H |n in terms of m|H |n
MIDTERM EXAM 2
PHYS 851 Quantum Mechanics I, Fall 2007
2
1. A system is described by the Hamiltonian H = 22 AA + A A AA A A , where A and A are the harmonic oscillator lowering and raising operators, which satisfy [A, A ] = 1. (a) [10 pts] Derive the Heis
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MIDTERM EXAM 1
PHYS851 Quantum Mechanics I, Fall 2007
1. [30 pts] BRAs, KETs, AND POSTULATES: The Hamiltonian of a physical system is H , and the eigenstates of H are |n , such that H |n = n |n . (a) [10 pts] Write an expression for the state of the
Double Slit Experiment
Consider a particle which passes through a single slit, then a double slit, and finally is detected by a screen:
a
s s
a
A1 (d )
d
Lecture 35: Path Integral formulation of quantum mechanics
P(d ) =
PHY851 Quantum Mechanics I Fall,
Lecture 33: Hilbert Space Transformations: The Heisenberg Picture
PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore
Real-Space transformations
Transformations in Real space include:
Translation Boost Rotation Parity reversal
We can treat them as either A
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 6:
1. The ne structure consists of the relativistic mass correction, the spin-orbit interaction, and the Darwin term. Based on the fact that j = + 1/2 or j = 1/2, show that for all n, , and j ,
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 6: SOLUTIONS
1. The ne structure consists of the relativistic mass correction, the spin-orbit interaction, and the Darwin term. Based on the fact that j = + 1/2 or j = 1/2, show that for all n,
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 7:
1. Spontaneous emission: In this problem, we will use the Fermi Golden Rule to estimate the spontaneous emission rate of an atom. We consider an atom which is initially excited, and therefor
NAME:
QUIZ 3
PHYS852 Quantum Mechanics II, Spring 2009 Hydrogen Fine Structure Quiz: a.) List the three eects which combine to give the ne structure, and give a brief description of each physical process. b.) Which of the three terms is responsible for se
NAME:
QUIZ 3
PHYS852 Quantum Mechanics II, Spring 2009 Hydrogen Fine Structure Quiz: a.) List the three eects which combine to give the ne structure, and give a brief description of each physical process. Answer: The three eects are: Relativistic Mass Eec
Phys 852, Quantum mechanics II, Spring 2008
Scattering theory
Prof. Michael G. Moore, Michigan State University
1
Statement of the Problem:
Scattering theory is essentially time-independent perturbation theory applied to the case of a continuous spectrum.
PHYS 852 Quantum Mechanics II
Spring Quarter 2009 Class Hours: MWF 10:20 AM to 11:10 AM Class Location: 1308 BPS Textbook: Quantum Mechanics, Volume two, Claude Cohen-Tannoudji Webpage: http:/www.pa.msu.edu/mmoore/852.html Instructor: Prof. Michael Moore