Final Exam
Physics 204B, 2013
Do all questions. They count equally, but I expect the early ones to be easier.
1. Three particles are in 6 states that have equal energy (and hence equal probability of
occurring).
a) Calculate the number of ways for the 3 p
Problem Set 4
Physics 204B
Due Monday February 3, 2014
Late HW accepted until class February 5
Do Riley 24.21, 25.22 and 25.23, plus the following:
a
x
a
1. Show that 0 (x+1)2 dx = sin a for 1 < a < 1. Note that the origin is a branch point,
and use a cur
Problem Set 2
Physics 204B
Due Wednesday January 22, 2014
Late HW accepted until class January 24
1. For the following functions, identify all singularities in the complex plane, their orders
(or that they are essential singularities), and the residue at
Problem Set 3
Physics 204B
Due in class Monday January 27, 2014
Late HW accepted until class January 29
1. Evaluate
cos
a) 0 1+xxdx
2 +x4
2 sin2 d
b) 0 5+3 cos
2. Show that:
/2
a) 0
d
a+sin2
=
2 a2 +a
(a > 0)
2
b) 0 sin 2 x dx =
x
2
3
c) sin 3 x dx =
Answers to Final Exam
Physics 204B, 2013
1. a)
(i) 6 possible states for each particle, so 63 = 216
(ii) You can imagine lining up the three bosons in a row and then inserting ve
dividers to signify where the boxes go, which gives 8 = 56.
5
(iii) Particle
Notes on Discrete Fourier Transforms
Physics 204B
Weve discussed Fourier series expansions, which apply to a periodic function or to a function
dened on a nite interval that can be assigned a periodic extension. The original function is
continuous, and to
Midterm Answers
Physics 204B, 2014
1. a) Numerator vanishes as z and denominator vanishes as z 2 , so the singularity has order
z 1 z 2 .
1
1. For the residue, ln(1z) (z21 z3 )2 z , so the residue is -1.
sin2 z
6
1
1
1
b) Expand the sine to get z sin
z 2
Final Exam
Physics 204B, 2014
1. (15 points)
Imagine a Schrdingers cat variant where two cats are in separate chambers
o
rigged with cyanide, with both chambers monitoring decays of the same radioactive sample
with a large number of atoms. In the rst cham
Final Exam Solutions
Physics 204B, 2014
1. The real question here is, if you know that there has been at least one decay in time t, what is
the probability that there have been at least 2 decays? The probability of having exactly n
n e
decays in time t fo
Midterm
Physics 204B, 2014
Do all problems. The four questions count equally.
1. Find the order of the specied singularity and its residue for each of these functions:
a)
ln(1z)
sin2 z
at z = 0
1
b) z 2 sin 1z at z = 1
2. The integral (t + 1) = 0 ez z t d