Problem Set 1
Physics 325A
Due Friday Jan 11, 2013
Note: All Griths problems are from Introduction to Quantum Mechanics 2nd Edition.
1. Ideal Fermi Gas
(a) Determine an expression for the mean energy per particle = E/N of an ideal Fermi
gas at T = 0 K in
Physics 325 EXAM 1 SOLUTIONS
Fri. Jan. 31, 2014
1. (40 pts) Fermi-systems at zero and nite temperatures
a) (10 pts) Explain the meaning of the Fermi-energy F in a system of Fermions at zerotemperature, and how you can calculate its value in a free-electro
Problem Set 3
Physics 325A
Due Friday Jan 24, 2014
Note: Short set due to holiday on Monday Jan 20.
1. Consider as an unperturbed system H 0 = (1/2)m 2 x2 , i.e., that of a simple harmonic oscillator with mass m and natural frequency , and a perturbation
Problem Set 4
Physics 325A
Due Friday Jan 31, 2014
Note: Short set due to Midterm Exam Fri Jan 31.
1
1. Calculate the rst order relativistic correction to the energy of a Hydrogen atom E0 for
the perturbation to the kinetic energy H 1 = p4 /8m3 c2 . To si
Problem Set 5
Physics 325A
Due Friday Feb. 7, 2014
1. Using the variational principle applied to the energy functional (in atomic units =
h
m = e2 /40 = 1),
E ( ) =
dx
1 d (x, )
2
dx
2
+ V (x)| (x, )|2 ,
with a normalized trial wave function (x, ) = exp(
Problem Set 1
Physics 325A
Due Friday Jan 10, 2013
1. Ideal Fermi Gas
(a) Determine an expression for the mean energy per electron = E/N of an ideal Fermi
gas at T = 0 K in terms of the electron number density n = N/V .
(b) Determine an expression for the
Problem Set 6
Physics 325A
Due Friday Feb. 14, 2014
1. Use the variational principle with a trial wave function (1, 2) = (r1 ) (r2 )0 , where
(r) = ( 3 / )1/2 er (in atomic units), to estimate the ground state energy of a Be+
ion. Express your result in
Problem Set 2
Physics 325A
Due Friday Jan 17, 2014
1. Metals and Insulators
a) Explain why a material like Cu is metallic, with a half-lled conduction band.
b) Explain (using the one-dimensional case as an example) why materials with lled bands
of electro
Problem Set 8
Physics 325A
Due Friday Feb. 28, 2014
Note: Short set due to Exam Mon. Feb. 24.
1. Spontaneous Emission: Einstein A coecient
a) Show (by converting the sum over photon modes to an integral) that the energy density
per unit volume u( ) of the
Problem Set 9
Physics 325A
Due Friday Mar 7, 2014
1. Consider an adiabatic process where the strength of a function potential V = (x)
is varied.
a) Calculate the geometric phase 0 for for the ground state n = 0 when is doubled from
to 2.
b) Calculate the
Problem Set 7
Physics 325A
Due Fri. Feb. 21, 2014
Note: Short set due to Holiday Mon. Feb. 17 and 2nd Midterm Exam Mon. Feb. 24
1. Consider the WKB solution for a bouncing ball of mass m with energy E conned to
the space z > 0 in a gravitational potential
1. ( x) exp( x )
1
E ( x) dx 3 ( x) exp 2 x
2
2
2
dE ( )
0
d
This is the exact result from solving the Schdinger Equation.
0
1
0
0
0
0
2. E0 E0 0 H ( 0) H ' 0 0 H 0 E gs
3. a. (r , )
# QED
3
exp( r )
0
0
T ( ) 2 3 r 2 exp( r ) 2 exp( r )dr 2 4 2r r
(
1. a. En1) n H ' n n ( x) 2 H ' ( x)dx 0 by the integral of an even and an odd function.
Note you can also apply the perturbation in terms of raising and lowering operators,
which will result in two orthogonal inner products, which is also zero.
n H' m
Problem Set 2
Physics 325A
Due Friday Jan 18, 2013
Note: All Griths problems are from Introduction to Quantum Mechanics 2nd Edition.
1. Metals and Insulators
a) Explain the nature of the conduction band in Na metal, e.g., why it is half-lled and
non-magne
Problem Set 3
Physics 325A
Due Friday Jan 25, 2013
Note: All Griths problems are from Introduction to Quantum Mechanics 2nd Edition.
Note: Short set due to holiday on Monday Jan 21.
1. Consider as an unperturbed system H 0 a simple harmonic oscillator wit
Problem Set 4
Physics 325A
Due Monday Feb. 4, 2013
Note: All Griths problems are from Introduction to Quantum Mechanics 2nd Edition.
Note: Changed due date due to Midterm Exam Friday Feb. 1.
1. Griths, Prob. 6.9 but with H11 = V0 (1 + ) and H22 = V0 (1 )
Problem Set 5
Physics 325A
Due Friday Feb. 8, 2013
Note: Short set due to Midterm Exam on Fri. Feb 1.
1. Using the variational principle applied to the energy functional (in atomic units =
h
2
m = e /40 = 1),
E ( ) =
dx
1 d (x, )
2
dx
2
+ V (x)| (x, )|2
d
Problem Set 6
Physics 325A
Due Friday Feb. 15, 2013
1. Use the variational principle with a trial wave function (1, 2) = (r1 ) (r2 )0 , where
(r) = ( 3 / )1/2 er (in atomic units), to estimate the ground state energy of a Li+ ion.
Express your result in
Problem Set 7
Physics 325A
Due Mon. Feb. 25, 2013
Note: Short set due to Holiday Mon. Feb. 18 and 2nd Midterm Exam Fri. Feb. 22
1. Consider the WKB solution for a bouncing ball of mass m with energy E in a gravitational potential V (z ) = mgz .
a) Estimat
Problem Set 8
Physics 325A
Due Friday Mar. 1, 2013
Note: Short set due to Exam Fri. Feb. 22.
1. Spontaneous Emission: Einstein A coecient
a) Determine the density of modes of electromagnetic radiation per unit volume g ( ) in
free space g ( ) = (2/V ) k (
Problem Set 9
Physics 325A
Due Friday Mar 8, 2013
1. Consider a particle initially in state n of a one-dimensional square well of length a
subject to a potential V = V0 sin(t/T ) for t > 0.
a) Solve the exact TD equations for the amplitude cm (t) to nd th
Problem Set 10
Physics 325A
Due Friday Mar 15, 2013
Note: Last Problem Set !
1. Greens functions
ik|r r |
m
a) Using 2 (1/r) = (4 ) (r) show that the Greens function G(r, r , E ) = 2h2 e |rr |
satises an inhomogeneous Schrodinger equation (E H0 )G(r, r ,
Please note: I accidentally graded this pset out of 12 points. In order to make it out of 10 points, I took the scores written
on each pset and multiplied it by 5/6 (but did not write the scaled score on each pset)
325: Pset 1 Solution
Problem 2
Part A