PHY491, Problem 1: Atoms (DUE: Fri 9/18/09)
GROUP:
(Dated: September 4, 2009)
I.
EXCITONS IN QUANTUM WELLS
A quantum well is a semiconductor nanostructure in which carriers are confined in a plane. A photon absorbed by the semiconductor quantum well crea
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 5
1. Show that the vector potential of a uniform magnetic eld B can be written as A(r) =
1 r B. Find
A. (3 pt) Consider a particle with charge q and mass m in a static
2
electrom
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 3
1. Consider a hydrogen atom. Seek the variational electron wave function in the form (r) =
(a3 )1/2 exp(r/a) with an unknown a (the function should be normalized!). Find the
ener
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 2
1. Two particles, with masses m1 and m2 and with charges e and e, are conned to a plane.
Assume that the center of mass is at rest and nd the energy spectrum of this twodimensio
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
YOUR NAME:
PHY 491  2013
Midterm Exam
1. In the intergalactic medium, oxygen (Z = 8) can be highly ionized. This has been seen in
farultraviolet spectroscopy. Consider an oxygen ion where only one electron is left. What
is the energy of a photon that it
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 1
1. Find the ground state energy of a positronium atom, a system that reminds a hydrogen atom
except that the proton is replaced with a positron. Express the energy in electronvo
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 6
1. Calculate the ground state energy of the hydrogen atom using the variational wave function of
the form (r) = A exp(r2 /a2 ) with one variational parameter a; the normalization
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 7
1. Chapter 4, problem 6 (6 pt)
2. Chapter 4, problem 4(a) (4 pt)
3. Chapter 5, problem 1 (a) and (b) (4 pt)
4. Consider a diatomic molecule with nuclear masses M and the typical
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 11
1. Consider a onedimensional lattice with lattice constant a. Make a plot of the energy Em k
in the reducedzone picture (projection on the rst Brillouin zone). Disregard the p
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 12, The Last and The Easiest
1. For a onedimensional crystal in the tightbinding approximation, assuming that the energy
dispersion law is Ek = E0 cos ka, nd the density of level
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 10
1. Show that, for a smooth function f (x) and for 0,
dx =
dx E f (x)
1
E 2 <f (x)<E + 1
2
Also use the denition of an integral to show that, for k = m/L with integer m and for
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 9
1. In elastic neutron scattering, a ux of neutrons is sent onto a crystal and the scattered beam
is observed for the neutrons with the same energy as in the incident beam. What i
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 8
1. Consider a twodimensional (2D) Bravais lattice with primitive vectors a1 and a2 . Find
(primitive) reciprocal lattice vectors. (3 pt) Find the area of the primitive cell of t
Atomic, Molecular, and Condensed Matter Physics Concepts in Physics
PHY 491

Fall 2013
PHY 491  2013
Atomic, Molecular, and Condensed Matter Physics
Problem Set 4
1. Find the terms for the electron conguration p3 (10 pt)
2. Find the terms for the conguration p4 . Hint: Start from p6 and think of subtracting
electrons; use the results obtai
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, September 5, 2012
Homework 1 Solution
1.1. Using a hydrogenic model, estimate the 1st ionization energy of a Li atom, assuming that
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, September 12, 2012
Homework 2 Solution
2.1. Calculate the ground state energy of a hydrogen atom using the variational principle. A
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Practice Exam 2
Student Name:
Common quantities:
Quantity
Name
1
A
1 aB
1 eV
1 Ry
1 Ha
1 amu
h
kB
NA
e
m
Angstrom
Bohr radius
electronVolt
Rydberg
Hart
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Practice Exam 2 Solution
Student Name:
Common quantities:
Quantity
Name
1
A
1 aB
1 eV
1 Ry
1 Ha
1 amu
h
kB
NA
e
m
Angstrom
Bohr radius
electronVolt
Ryd
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Practice Exam 1
Student Name:
Useful Formulas:
Magnetization M of a multiplet with a given total angular momentum J and Land g factor gJ
e
is given by
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Practice Exam 1 Solution
Student Name:
Useful Formulas:
Magnetization M of a multiplet with a given total angular momentum J and Land g factor gJ
e
is
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, November 21, 2012
Homework 10 Solution
10.1. Show that for a diatomic chain (two dierent masses M1 and M2 that interact with same f
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, November 14, 2012
Homework 9 Solution
9.1. This problem on the cohesive energy of bcc and fcc neon compares the subtle dierence bet
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, October 31, 2012
Homework 8 equivalent to Practice Exam 2 Solution
Common quantities:
Quantity
Name
1
A
1 aB
1 eV
1 Ry
1 Ha
1 amu
h
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, October 24, 2012
Homework 7 Solution
7.1. Use the following equation for the drift velocity of an electron in a constant electric e
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, October 17, 2012
Homework 6 Solution
6.1.
(i) Calculate the density of states of the electron gas in 2 and 1 dimensions.
(ii) Deriv
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, October 10, 2012
Homework 5 Solution
5.1. In each of the following cases indicate whether the structure is a primitive Bravais latt
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, September 26, 2012
Homework 4 Solution
4.1. Consider an atom with the 3 S1 ground state. What is the value of Land gfactor? Find t
PHY 491: Atomic, Molecular, and Condensed Matter Physics
Michigan State University, Fall Semester 2012
Solve by: Wednesday, September 19, 2012
Homework 3 Solution
3.1. Consider an openshell atom with 4 electrons in the pshell (p4 ), such as the oxygen a