PHYS 424
Spring 2009
Homework Assignments
HW1
Remember: You need not submit your homework solution. Instead, one of these problems,
or a part of it, will be on the quiz on February 9. Time will be limited, so that you will
have little chance of completing the quiz, if you have not worked out the problem before.
1.2.
1.3.
1.4.
1.9.
1.15.
1.18.
As part (c) also calculate the probability that the ball has fallen more than h/3. Is
this probability 50%? What is the median of the distance traveled?
Hint: Notice the useful integrals in the inside back cover of your book.
Start with sketching the wave function.
HW2
Quiz on February 16.
2.1. a and c
2.2.
2.4.
2.7.
2.36.
2.38
2.10
2.11
HW3
One of these problems, or a part of it, will be on the quiz of February 23.
2.12
2.15
2.16
1.14 and 2.19
2.49 Use the compact form x,t   A exp[ f (x,t)], when you do not need all the
details of the function. Here A  m   and   A 2 . In part (b) express every
trigonometric function with functions of argument t. In (c) the
y = x-a cos(t)
 substitution might work.
14


B
In quantum mechanics, the classical quantity xp is represented with the
symmetrized operator Qˆ  12 ( xˆpˆ  pˆ xˆ ). (a) Express operator Q in terms of raising
and lowering operators. (b) Calculate the expectation value of Q for the n-th
eigenstate of the harmonic oscillator.
 functions and their squares for the n = 5 and n = 6 eigenstates of
Optional: Plot the wave
the harmonic oscillator. Do not worry about the normalization, the objective is to
see the shape of the curves. Mark the classically allowed region on your graphs.
(Use a computer program – EXCEL, MATLAB, Mathematica, Origin, etc. – to
prepare the graphs. You can make markings by hand.)
HW4
One of these problems, or a part of it, will be on the test of March 2.
2.21
2.23 (b) and (c);
2.24 (b);
2.26.
2.35
2.40 (a)
C
The eigenfunction of the bound state of a particle in the potential V(x) (x)
is (x)  e  x , where   m / 2 . (a) Calculate the expectation value of x2 and
the expectation value of the potential energy for this state. (b) The energy
eigenvalue of the bound state is E  m 2 /2 2 . How does
 your result compare to
this value?


HW5

One of these problems, or a part of it, will be on the quiz of March 9.
D. A particle is in an infinite square well defined by the potential
0,
if  a2  x  a2
V x   
else
,
Its wave function at t = 0 is
 2n 1 
2m 
x,0  Acos
x  sin 
x ,

 a

  a

where n, m = 1, 2, 3,…
(a) Sketch the wave function for the n = 2, m = 1 case.
(b) Determine A.
 is found, if the energy of the particle is measured on an ensemble of identical
(c) What
systems? (Give possible measured values and their probabilities.)
E. The initial state of a particle is described by the wave function
Ax a  x , if 0  x  a
x,0  
else
 0,
(a) Is this particle moving to the right or to the left at t=0 ? Calculate <p>.
(b) Construct a wave function 1 x,0 that describes a particle moving to the left.

Problems from the book:

3.2
3.3
3.4 (b) and (d)
Recommended practice problem: Sketch the following functions without using a
computer or calculator. Try to combine known functions, find zeros and other special
points and features, rather than calculating a set of substitution values.
 x 
 x 
(a)
y  tan  cot , for  3  x  5
2 
2 
1 x
(b)
y
, for  5  x  5
1 x
 1 
y  1 ln( x), for 0  x  
 (c)
 x 

HW6

One of these problems, or a part of it, will be on the quiz of March 23.
3.5 (a) and (c)
3.6
3.7
3.10
3.11
3.13
3.14
3.15
3.17
3.22
HW7
One of these problems, or a part of it, will be on the quiz of March 30.
4.2
4.3
4.5 but only Yll for arbitrary l.
4.19

4.24
4.38 First assume an asymmetric oscillator with
1
V (r)  m  x2 x 2   y2 y 2   z2z 2 
2
and find the energies and wave functions for this case. Than make the three
angular frequencies equal to get eq. (4.188) and answer the original questions of
the problem.

HW8
One of these problems, or a part of it, will be on the quiz of April 6.
4.10 Do R30 only
4.13
4.14 (Optional: Do the same for the n=2, l=0 and the n=2, l=1 states)
4.18
4.22
B Sketch R42, R50, and R52 for the hydrogen atom. The figure need not be to scale, but it
must show the proper behavior at the origin and in infinity and must have the
correct number of zeros.
HW9
One of these problems, or a part of it, will be on the quiz of April 13.
4.28
4.29
4.34
A The z component of the spin of a spin-1/2 particle is measured and the result /2 is
obtained. Soon after, Sv  4Sx  3Sz /5 is measured. What are the possible results
and their probabilities? Finally S2 is measured. What are the possible values and their
probabilities?


B A beam of spin-one
particles in the  (z0 ) (m=0) state arrives at a Stern-Gerlach
apparatus that measures the x component of each particle. What are the possible
results of the measurement and their probabilities? Use
0
0 1 0



(z )
 0  1 and Sx 
1 0 1

2 
 

0
0 1 0

C A series of measurements is carried out on an unpolarized beam of heavy particles.
They carry no electric charge, but they have spin ½ and an associated magnetic
moment. First the beam is passed through a Stern-Gerlach apparatus that splits it into
two sub-beams, one with the x-component of the spin in the +x direction, the other
with spin in the –x direction. The beam of particles with negative spin is passed
through a second Stern-Gerlach apparatus that splits it into two beams, one with spin
in the +z direction, the other with spin in the –z direction. What fraction of the
particles is found in the beam with spin in the +z direction? These particles are
analyzed with a third Stern-Gerlach apparatus. What will be the result of the
measurement, if this apparatus measures spin in the x-direction? What if the third
Stern-Gerlach apparatus measures spin in the z-direction?
D Now the beam of particles in the previous experiment is first passed through a SternGerlach apparatus that splits it into two sub-beams, one with the x-component of the
spin in the +x direction, the other with spin in the –x direction. The beam of particles
with positive spin is passed through a region where they are in a uniform magnetic
field in the z-direction for a time interval T. (a) Determine the spinor describing these
particles when they leave the region. (b) The particles are analyzed using a SternGerlach apparatus that measures the x-component of the spin. What will be the result
of the measurement? (c) What if this Stern-Gerlach apparatus measures the zcomponent of the spin?
Optional: 4.33. (This is a really fun problem with illuminating result. But the **
designation is well warranted.)
HW10
One of these problems, or a part of it, will be on the quiz of April 27.
We will cover a few selected sections during the rest of the semester. I will give closer
guidance on what to read and to what depth. For this week:
Reading: 5.1 in depth,
5.2 in preparation for next week
5.3-4: skip
6.1 in depth
Problems:
5.2 (a)-(c)
5.4
5.7
5.32
6.1
6.2
HW11
One of these problems, or a part of it, will be on the quiz of May 4.
Reading: 5.2 in detail
6.2-3 in depth
6.4 for the basic physics involved
6.5 optional
Problems (long and hard ones):
6.8
6.14
6.36
HW12
One of these problems, or a part of it, will be on the quiz of May 11.
Reading: 7.1 in detail
7.2 focus on pp. 302-3
9.1 in detail
9.2.1 and 9.2.2 in detail
9.2.3-9.3.2 read
9.3.3 in detail

7.1 (b)
7.3
F An electron is in the strong static field B0 that is parallel to the z-axis. A much weaker
constant field B’ in the x-direction is turned on at t = t0 and off at t = tf. How will this
field change the state of the electron?
H A CO molecule is described as a 1-d harmonic oscillator with potential energy
1
V0 x   kx 2 and mass m. (a) What is the exact meaning of x and m in this case? (b)
2
Describe the vibrational energy levels and the corresponding optical spectrum of CO
gas. (Neglect transitions involving any change of electronic and/or rotational states.
You can make reference to the selection rule determined in a homework problem or
work it out again.) (c) Especially at larger amplitudes, the anharmonicity of the
potential becomes important; describe it with the perturbing potential V' x  x 4 .
Calculate the change of the energy levels to first order. (d) How does the
anharmonicity affect the measured vibrational spectrum?
