PHY-852 QUANTUM MECHANICS II
Homework 6, 40 points
February 23 - March 2, 2011
Identical particles
Reading: Quantum Physics, Volume 2, Sections 15.1-4, 17.1-4, 18.1-3
1. /10/ a. Consider a system of two identical particles of arbitrary spin s.
Find the nu
PHY-852 QUANTUM MECHANICS II
Homework 2, 40 points
January 26 - February 2, 2011
Klein-Gordon-Fock equation. Spin 1/2
Reading: Quantum Physics: Volume 2, Section 11.10;
Volume 1, Sections 16.1-16.4, 20.1-20.3
1. /12/ a. In the Problem 11.1 (Volume 2) it i
PHY-852 QUANTUM MECHANICS II
Homework 1, 40 points
January 19 - 26, 2011
Klein-Gordon-Fock equation (KGFE)
Reading: Quantum Physics, Volume 2, Sections 11.1-11.10.
1. /10/ a. Consider a particle of mass m and electric charge e moving in
an electrostatic p
Variational method and diagonalization
Take a complete orthonormalized set of states |ni. Select N states of this
set, |1i, |2i, ., |N i, and look for the approximate solution of the stationary
Schr
odinger equation,
= E,
H
(1)
in the form of the superpo
PHY-852: QUANTUM MECHANICS II
Quiz 5
March 18, 2011
NAME.
PROBLEM. Consider a neutron in the gravitational field near the surface
of the Earth. Estimate the energy of its quantum ground state using the variational method with a reasonable (and simple enou
Phys 852, Quantum mechanics II, Spring 2009
Time-Independent Perturbation Theory
Prof. Michael G. Moore, Michigan State University
1
The central problem in time-independent perturbation theory:
Let H0 be the unperturbed (a.k.a. background or bare) Hamilto
PHYS852 Quantum Mechanics II, Spring 2010
HOMEWORK ASSIGNMENT 9: Solutions
Topics covered: hydrogen hyper-ne structure, Wigner-Ekert theorem, Zeeman eect
1. Relations between V and J : For a rotation by about the z-axis, we have U Vz U = Vz , U Vx U =
cos
PHYS852 Quantum Mechanics II, Spring 2010
HOMEWORK ASSIGNMENT 8: Solutions
Topics covered: hydrogen ne structure
1. [10 pts] Let the Hamiltonian H depend on the parameter , so that H = H (). The eigenstates and
eigenvalues of H are then also functions of
PHYS852 Quantum Mechanics II, Spring 2010
HOMEWORK ASSIGNMENT 8
Topics covered: hydrogen ne structure
1. [10 pts] Let the hamiltonian H depend on the parameter , so that H = H (). The eigenstates and
eigenvalues of H are then also functions of , i.e. En =
PHYS852 Quantum Mechanics II, Spring 2010
HOMEWORK ASSIGNMENT 7: Solutions
Topics covered: addition of three angular momenta, degenerate perturbation theory
1. Consider a system of two spin-1/2 particles, described by S1 and S2 , and one spin-1 particle,
PHYS852 Quantum Mechanics II, Spring 2010
HOMEWORK ASSIGNMENT 7
Topics covered: addition of three angular momenta, degenerate perturbation theory
1. Consider a system of two spin-1/2 particles, described by S1 and S2 , and one spin-1 particle, described
b
PHY-852 QUANTUM MECHANICS II
Homework 4, 40 points
February 9 - 16, 2011
Symmetries
Reading: Quantum Physics, Volume 1, Sections 7.10, 8.1-5, 16.1-16.4, 20.5-6
1. /32/ Consider the set of physical quantities describing the motion of a
spinless charged par
PHY-852 QUANTUM MECHANICS II
Midterm test
March 4, 2011
Total 30 points
NAME.
1. Write down the Dirac equation for a particle moving in a spherically symmetric electrostatic potential (r). List all constants of motion.
2. Consider a system of two particle
PHY-852: QUANTUM MECHANICS II
Quiz 2
February 4, 2011
NAME.
PROBLEM. For the following operators find out if they correspond to conserved quantities for free motion of a relativistic particle of spin 1/2:
2
a. orbital momentum squared ~` ,
b. spin squared
PHY-852 QUANTUM MECHANICS II
Homework 7, 40 points
March 14 - 23, 2011
Discrete spectrum and variational method
Reading: Quantum Physics, Volume 1, Sections
3.1-3; 3.5; 5.6; 10.1-3; 11.1; 17.1-4
1. /6/ For a particle bound in the potential box (0 < x < a)
PHY-852 QUANTUM MECHANICS II
Homework 5, 40 points
February 16 - 23, 2011
Symmetries and spins
Reading: Quantum Physics, Volume 1, Sections 8.3-5, 16.1-10, 20.2-3
1. /8/ A non-relativistic neutron (spin magnetic moment ) is moving in an
axially symmetric
PHY-852: QUANTUM MECHANICS II
Quiz 7
April 1, 2011
NAME.
PROBLEM. A particle of mass m and electric charge e is placed in the onedimensional harmonic oscillator potential of frequency . The uniform electric
field E is applied along the same axis.
a. Find
PHY-852: QUANTUM MECHANICS II
Quiz 6
March 25, 2011
NAME.
PROBLEM. A quantum harmonic oscillator with mass m and elastic force
kx is perturbed by a small change k k + . Using perturbation theory
find the change of the energy spectrum and compare the resul
PHY-852 QUANTUM MECHANICS II
Homework 9, 40 points
March 30 - April 6, 2011
Stationary perturbations
Reading: Quantum Physics
Volume 1, Sections 4.7, 16.6-16.7, 17.6, 19.1-19.4, 24.1-24.4
1. /12/ An electron is confined to a circle of radius R in the xy-p
PHY-852: QUANTUM MECHANICS II
Quiz 3
February 11, 2011
NAME.
PROBLEM. Consider two Hamiltonians of a non-relativistic particle of
spin 1/2 which consist of kinetic energy, spin-independent central potential and
different spin-dependent potentials,
a.
2
1
PHY-852: QUANTUM MECHANICS II
Quiz 1
January 28, 2011
NAME.
PROBLEM. a. Calculate the mixed product of three Pauli matrices
A = ~ [~ ~ ] .
(1)
b. Reduce to the form linear with respect to the Pauli matrices the operator
k = (a ~ )k ,
B
(2)
where a is a nu
PHY-852: QUANTUM MECHANICS II
Quiz 4
February 18, 2011
NAME.
PROBLEM. A proton state is characterized by the momentum p along the
z-axis and by the certain value h = +1 of the helicity [projection h = (~ p)/|p|
of the proton spin (s = (1/2)~ ) on the dire
PHYS852 Quantum Mechanics II, Spring 2010
HOMEWORK ASSIGNMENT 6: Solutions
Topics covered: Time-independent perturbation theory.
1. [30] Two-Level System: Consider the system described by H = Sz + Sx , with > 0, where
Sx and Sz are components of the spin
PHYS852 Quantum Mechanics II, Spring 2010
HOMEWORK ASSIGNMENT 6
Topics covered: Time-independent perturbation theory.
1. [30] Two-Level System: Consider the system described by H = Sz + Sx , with > 0, where
Sx and Sz are components of the spin vector of a
QUANTUM MECHANICS SUBJECT EXAM Spring 2008 ID NUMBER:
1. POSTULATES OF QUANTUM MECHANICS: A certain quantum system is governed by th energy eigenstate of the Hamiltonian H = n=0 En |n n|, where the state |n is the n the system, with eigenvalue En . Assume
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