Solutions to Homework 4
1
Physics 581, Spring 2007
Counting States of Particles with Dierent Statistics
(a) 1 Conguration
One conguration with each fermion occupying a dierent state.
(b) 10 Congurations
There is one conguration with all bosons in dierent
Solutions to Homework 1
1
Phys581, Spring 2007
Harmonic Oscillator
1)
H0 = a a +
1
2
2)
Remember that:
a |n
a |n
n |n 1
=
n + 1 |n + 1
=
from witch we get:
n| a |m =
m + 1n,m+1
And by iterating this:
n + m| am |n
=
(n + m)!
n!
n m| am |n
=
n!
(n m)!
3)
It
Solutions to Homework 2
1
Phys581, Spring 2007
Charged Particle on a Plane
1)
The canonical momentum:
p=
rL
e
= mr + A(r)
c
2)
H
1 2 e
= r p mr e A(r)
2
c
p e A(r) 1 mr2
=r
c
2
1 2
=
mr
2
2
e
1
p A(e)
=
2m
c
3)
We expand the Hamiltonian:
H=
12
e2
e
p+
p
Solutions to Homework 3
1
Phys581, Spring 2007
Time evolution of a spin an an external magnetic eld and Spin
Resonance
(1)
dSi (t)
dt
=
=
=
1
Si (t), H (t)
i
ge 1
Bj (t) Si (t), Sj (t)
2mc i j
ge 1
2mc i
Bj (t)i
ijk Sk (t)
j
And in vector form:
ge
dS (t)
1
Quantization of the Electromagnetic Field
1.
The electric and magnetic eld are given by the vector potential:
E
=
B
=
1 A
c t
A
The electromagnetic energy reads:
H=
1 1
d3 r
8 c2
A
t
2
+
A
2
The time dependence of the vector potential can be restored by