Heat capacities in a magnetic system
The reasoning here parallels the reasoning connecting CV with Cp in uid systems. There are three main
parts:
A: Begin with the known master relation for E(S, H):
dE = T dS M dH.
Apply a Legendre transformation to varia
Stumbling in the thermodynamic dance
a. Attempting to take a step in the thermodynamic dance, we try a Legendre transformation to variables
T , p, and by dening
= G N.
But = G/N , so = 0 and any attempt to say, for example,
= N,
T,p
is bound to fail. The
Heat capacity at constant pressure
The argument for CV was: By denition,
CV (T, V, N ) = T
S
T
.
V,N
But the energy dierential is
dE = T dS p dV + dN.
To reect the constant V and N in the denition of CV above, restrict this equation for dE to changes at
c
Fluid work
initial
32 Pa
b.
c.
p
a.
d.
1 Pa
1 m3
0
final
8 m3
V
In general we have
nal
Work =
p(V ) dV,
initial
but along path a
p(V ) =
K
V
K = pi Vi = pf Vf .
where
Thus the work along path a is
Vf
Work =
Vi
K
K
1
dV =
V
( 1) V 1
Vf
=
Vi
1
K
1
1
( 1) V
Quantum Mechanics 2011
Model Solutions for Second Exam
1. Commutator
Apply
[A, B] = AB B A = cB A
to the ket |a which has A|a = a|a . We nd
AB|a B A|a = cB A|a
AB|a B(a|a ) = cB(a|a )
A(B|a ) a(B|a ) = ca(B|a )
A(B|a ) = a(B|a ) + ca(B|a )
A(B|a )
Ladder Operators for the Simple Harmonic Oscillator
a. Simple algeba shows that
h
( + a )
a
2m
m
h
( a )
a
p = i
2
x
=
b. Matrix elements.
m|n
a
=
=
m| |n
a
n m,n1
n + 1 m,n+1
h
=
n m,n1 + n + 1 m,n+1
2m
m
h
n m,n1 n + 1 m,n+1
= i
2
m|n
x
m|n
p
[Ques
Square Well with a Bump
V (x) =
0
2
p
H0 =
+ V ()
x
2M
H = V ()
x
where
0
V (x) =
V
0
where
L < x
0 < x < L
x < 0
(L + a)/2
(L a)/2
< x
< x <
x <
(L + a)/2
(L a)/2
The rst-order energy corrections are
(1)
En
=
n |H |n
(L+a)/2
=
=
=
=
=
=
=
n (x) V n (x
Angular Momentum
Angular momentum trivia
a. Assume that A commutes with Lx and Ly :
[A, Lx ] = 0,
[A, Ly ] = 0.
(1)
Then it follows from
[Lx , Ly ] = i Lz ,
h
that
[A, Lz ]
1
[A, [Lx , Ly ]
[Use the Jacobi idenity. . . ]
i
h
1
=
[Ly , [A, Lx
Anharmonic Oscillator
a. Using the results from the problem Ladder Operators for the Simple Harmonic Oscillator,
m|3 |n
x
=
=
m|
x
h
2m
|2 |n
x
3/2
m,
=
h
2m
3/2
+
+ 1 m,
n(n 1)
,n2
+1
+ (2n + 1)
,n
+
(n + 1)(n + 2)
,n+2
m,
+
h
2m
1
3/2
1
n(n 1)
,n2
+
Quantum Mechanics
Model Solutions for Sample Exam for Second Examination
1. The state evolves in time to
h
h
h
|(t) = 2 e(i/ )E2 t |2 + 6 e(i/ )E6 t |6 + 8 e(i/ )E8 t |8 ,
so it remains always a linear combination of even energy eigenstates. Now the momen
Degenerate Pertubation Theory in a Two-State System
a. Using the initial basis (call it cfw_|1init , |2init ):
1init |H |1init
=
1
0
a3
a1
a1
a3
1
0
= a3
2init |H |2init
=
0
1
a3
a1
a1
a3
0
1
= a3
Using the second basis (call it cfw_|1second , |2second ):
Expressions for SHO Ladder Operators
The lowering operator a acts upon energy eigenstate |n as
a|n =
n |n 1 .
Since we know how a acts upon every element of a basis, we know how it acts upon any state.
The outer product expression
m |m 1 m|
m=0
similarly
Quantum Mechanics
Sample Exam for Second Examination
1 A one-dimensional simple harmonic oscillator has initial state
| = 2 |2 + 6 |6 + 8 |8 .
What is the expected momentum p , and how does it change with time?
2 Evaluate mcfw_ xp for any state | . ( mcf
Oberlin College Physics 411, Electrodynamics, Spring 2014
Assignment 4
Friday, 18 April
Reading: Within Griths chapter 10 on Potentials, read sections 10.1 and 10.2.
Problems: Due Friday, 25 April.
Griths 9.12: Time averages
Griths 9.13: Maxwell stress
Group velocity problems
Griths Electrodynamics, fourth edition, problem 9.23: Water waves and quantal waves
(a) Deep water waves
v
=
v
=
=
vg
2
k
2
=
k
k
2k
d
1
2
1
=
= v
dk
2
k
2
=
=
(b) Quantum mechanical waves
h
The form of the wave Aei(pxEt)/ , co
Time averages
Griths, Electrodynamics, fourth edition, problem 9.12
We have
f (r, t) = A cos(k r t + a ) = ecfw_Aei(krt+a ) = ecfw_f ei(krt)
where f = Aeia . And we have
g(r, t) = B cos(k r t + b ) = ecfw_Bei(krt+b ) =
ecfw_ei(krt)
g
where g = Beib .
Maxwell stress tensor for light
Griths, Electrodynamics, fourth edition, problem 9.13
The wave in question is (see Griths equation 9.48)
E(z, t)
=
B(z, t)
=
E0 cos(kz t + ) Eg (z, t)
x
x
1
1
E0 cos(kz t + ) = Eg (z, t)
y
y
c
c
(The subscript g stands for
Physics 410
file "hw-04"
Assignment #4
Applications of the Canonical Ensemble
Fall 2013
September 29, 2013
Assignment #4
PHYS-410
Fall 2013
Mr. Scofield
Reading
Topics this week (in Ch. 6) include density of states and Maxwell-Boltzmann
distribution. Plea
Physics 410
file "hw-03.doc"
Assignment #3
The Canonical Ensemble (Boltzmann Factor)
Fall 2013
September 20, 2013
Assignment #3
PHYS-410
Fall 2013
Mr. Scofield
Announcements
1. Your solutions to this HW assignment are due on Friday, September 27.
Reading
Physics 410
file "hw-02"
Assignment #2
Introduction to Quantum Stat. Mech.
Fall 2013
September 13, 2013
Assignment #2
PHYS-410
Fall 2013
Mr. Scofield
Announcements
1. The solutions to your first homework assignment have been distributed in class.
Reading
Physics 410
file "hw-01"
Assignment #1
Classical Thermodynamics
Fall 2013
September 3, 2013
Assignment #1
PHYS-410
Fall 2013
Mr. Scofield
Announcements
1. Please fill out and return to me the student information sheet. I need this at the end of our
first
Physics 410
file "hw-06"
Assignment #6
Fall 2013
October 21, 2013
Assignment #6
PHYS-410
Fall 2013
Mr. Scofield
Announcements
Reading
this next week you will stay in Chapter 5. Review Sections 1-3. Skim through the rest of
the chapter, reading portions th
Physics 410
file "hw-08"
Assignment #8
Ideal Fermi Gas and Blackbody
Fall 2013
November 8, 2013
Assignment #8
PHYS-410
Fall 2013
Mr. Scofield
Announcements
Reading
Read the rest of Chapter 7, with particular emphasis on section 7.4 (blackbody radiation)
a
Physics 410
file "hw-07"
Assignment #7
Quantum Distribution Functions
Fall 2013
November 1, 2013
Assignment #7
PHYS-410
Fall 2013
Mr. Scofield
Announcements
Reading
Begin reading Chapter 7 of the textbook. In treating systems of N, indistinguishable
parti
PHYS-410
Statistical Mechanics
Fall 2013
Exam 2 Equation Sheet
Thermodynamic Potentials
Microcanonical Ensemble:
1
p
Enthalpy, H S , N , p U pV
dS dU dV dN
T
T
T
Helholtz Free Energy, F V , N , T U TS
S k B log
Gibbs Free Energy, G N , T , p F pV
log
Physics 410
file "hw-05"
Assignment #5
The Canonical Ensemble (Boltzmann Factor) II
Fall 2013
October 12, 2013
Assignment #5
PHYS-410
Fall 2013
Mr. Scofield
Announcements
Reading
If you have not already done so please read the notes I have posted online r
Phys-410
Mr. Scofield
Statistical Mechanics
Secon Exam Topic List
Fall 2013
P410 Second Exam Topic List
Maxwell-Boltzmann speed and velocity distributions
Mean speed, rms-speed, peak speed
Thermal deBroglie wavelength and its meaning
Free Energy (i.e. Hel
Phys-410
Mr. Scofield
Fall 2013
Statistical Mechanics
First Exam Topic List
P410 First Exam Topic List
Numerical values
Boltzmann constant
Mass of a proton, neutron, and electron
Basic properties of periodic table (H, He, C, N, O)
Avagadros number, Planck