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Physics 135b Winter Term, 2013
Cover Page
Final Examination
Due Date: Wednesday, March 20, at 5:00 pm SHARP to Room 12 E. Bridge
in the Phys135b Homework Box. Alternatively, you may send a pdf le of your
solutions to Nick Hunter-Jones, nickrhj@gmail.com
Ph135b Homework 1 Solutions
Nick Hunter-Jones
Winter 2013
Problem 1
We want to show that the Hamiltonian of a system of particles and an electromagnetic eld, given
by
1
1
1
2
Hrad =
m v +
d3 x 0 E 2 + B 2 ,
(1)
2
2
0
is a constant of the motion. We start
Ph135b Homework 2 Solutions
Nick Hunter-Jones
Winter 2013
Problem 1
A beam splitter transforms incoming modes ai and bi to outgoing modes ao and bo , where
and
bo = bi i 1 ai .
ao = ai i 1 bi
(1)
a) We want to show that the transformation is generated by
Ph135b Homework 3 Solutions
Nick Hunter-Jones
Winter 2013
Problem 1
We have operators S+ , S , and S3 , dened by
S+ = a b,
S = ab ,
1
S3 = (a a b b).
2
(1)
To show that these operators generate a representation of the SU (2) Lie algebra, we need to
show t
Ph135b Homework 4 Solutions
Nick Hunter-Jones
Winter 2013
Problem 1
We want to show that for a linear arrangment of n beam splitters, the noramlly ordered variance
V out () for the an+1 mode is related to that of the input eld V in () for the a1 mode by
n
Ph135b Homework 5 Solutions
Nick Hunter-Jones
Winter 2013
Problem 1
The master equation for the damped harmonic oscillator, a single mode of the electromagnetic eld,
is,
d
= i[a a, ] + ( + 1)(2aa a a a a) + n(2a a aa aa ),
n
(1)
dt
2
2
where is the decay
Ph135b Homework 6 Solutions
Nick Hunter-Jones
Winter 2013
Problem 1
a) Consider a eld which propagates through a medium whose amplitude transmission coecent
is given by t = t0 e0 . We assume that 0
1 and that the input state is a coherent state of
amplitu
Ph135b Homework 7 Solutions
Nick Hunter-Jones
Winter 2013
Problem 1
Consider a two-level atom of transition frequency 0 , trapped in a high Q cavity, where the cavity
has photon decay rate 2 and the atom has spontaneous emission rate 2 . If the atom inter