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PY354 Modern Physics
Final Exam
December, 16, 2002
Name:__________________
BU ID#_____________________
Section time:_____________
This is a closed book exam. Any formulas you are likely to need, and would have trouble
remembering are provided on the back page. Please do not use formulas or expressions
stored in your calculators. Please write all your work in the space provided, including
calculations and answers. Please circle answers wherever you can. If you need more
space, write on the back of these exam pages, and make a note so the grader can follow.
This is a long exam, and in all likelihood, you will not finish, so don't be upset by leaving
things incomplete. Please note the point totals for each problem and section as they will
guide you in how most effectively to apportion your time. There is a total of 112 points.
Good luck!!
Problem 1. (24 pts)
Consider the n=5 bound state of an infinite potential well of width L.
a.) (4 pts) Write down the wavefunction without worrying about the proper
normalization.
b.) (4 pts) How many nodes (maxima) does the wavefunction have across the well?
c.) (4 pts) How many nodes does the probability density for this state have across the
well?
d.) (4 pts) Determine the locations in the x direction of the nodes in the probability
density.
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e.) (8 pts) Derive a general formula for the location of the nodes in the probability density
for the bound states (any n) of an infinite potential well of width L.
Problem 2. (30 pts)
For this problem you will solve the time-independent
Schrodinger equation for a cylindrical quantum dot. The
potential is defined by a disk of radius a, and height h, as
shown in the figure to the right. Inside the disk, the
potential U is zero and outside the disk the potential U is
infinite. The Laplacian in cylindrical coordinates (r, , z)
is given by:
1     1 2
2
2 

r   2
r r  r  r  2 z 2
a
h
a.) (2 pts) Write down the time-independent Schrodinger equation for the interior of the
cylindrical quantum dot (U=0) in cylindrical coordinates.
b.) (6 pts) Start solving this problem by using the technique of separation of variables.
Define a total wavefunction as a product of three functions, each of r, and z. Then
plug-in and solve for an equation showing the independence of functions of these
variables.
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c.) (4pts) Find the general form of the solutions to the function of the height, z.
d.) (4 pts) Apply boundary conditions and find the energy eigenvalues and allowed
quantum numbers for the function of the height, z.
e.) (6 pts) Find the general form of the solutions to the function of angle, 
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f.) (4pts) Apply the boundary conditions for the angular variable and determine the
allowed values of the quantum number and energy eigenvalues.
g.) (4 pts) What physical property does the angular variable and its quantization
correspond to?
Problem 3. (16 pts)
The energy levels of a 3D quantum box, of equal length L on each side are given by:
 2 2 2
E nx ,n y ,nz 
n x  n y2  n z2 .
2mL2
a.) (4 pts) Write down the energy levels and degeneracy for the first 7 states.


b.) (6 pts) Now imagine a static, DC electric field is applied in the y-direction. Describe
and carefully justify physically, first in words, what happens to the energy levels.
Consider and show how the electric field would be written into the Schrodinger Equation,
and whether separation of variables would still apply, and if so, how. (Do not perform
any calculations)
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c.) (6 pts) Next, in the plot to the right, show the energy
level diagram both before and after application of the
electric field. Be careful about which levels change in
energy. The after plot will be only approximate as you
are not required to calculate any energy levels.
Energy
without with
field
field
Problem 4. (8pts)
The expression for Reflectivity across a barrier of width L and height U with an incident
wave of energy E is (E>U):
 2 m( E  U 0 ) 
sin 2 
L



.
R


2
m
(
E

U
)


E E
0

sin 2 
L  4
 1

U
U
0 
0



a.) (4 pts) Determine the condition for resonant transmission, that is the incident energies
for which all the incident particles are transmitted.
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b.) (4 pts) Imagine that you wanted to create a "mirror" for particles which would only
transmit a very particular incident energy such that all other near-by energies would be
effectively reflected. One way to accomplish this is to build a repeating series of barriers,
all the same width and height, such that the resonant condition is always met. Show by
sketch what happens to the transmission probability plotted as a function of (E/U) for
more and more barriers. Show by a sketch where the barriers should be located relative to
each other.
Problem 5. (10 pts)
a.) ( 4pts) Explain in words why it is necessary that the wavefunction of a many-particle
state of indistinguishable particles be either symmetric or anti-symmetric.
b.) (6 pts) Determine and write down the possible wavefunctions for the two (noninteracting) spin-one-half particles in an infinite potential well. Start by solving for the
possible spin-state configurations, and then use the symmetric and anti-symmetric twoparticle wavefunctions on the formula page to determine the total wavefunction.
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Problem 6. (16 pts) Consider the total angular momentum resulting from the sum of
individual angular momentum of two particles. One has L1=3, and the other has L2=2.
a.) (3 pts) Write down the vector form for the total angular momentum:
b.) (3 pts) Determine the values the total angular momentum can take, and for each,
find the possible values of z-component of total angular momentum
c.) (3 pts) Consider now the addition of angular momenta from two electrons. As in
b), write down the values the magnitude of total angular momenta and for each
the z-component(s).
d.) (4 pts) Now, by analogy with 5b) above, write down the two particle
wavefunctions for the four possible states.
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Problem 7. (10 pts)
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are in a harmonic potential. The
2
1
energy levels of the harmonic potential are given by: E n  (n  ) 0 ,
2
n  0,1,2,3,... Determine the minimum possible total energy.
a.) (6 pts) Six (non-interacting) particles of spin s 
c.) (4 pts) What would the total energy be if they were spin 1?
Semiconuctor Physics Question:
Band structure, why are there gaps? Conceptual.
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Formulas for PY354 Final Exam, Dec 16th, 2002
E  hf  
p
h
KEmax  hf  
 k

 2  2  ( x, t )
 ( x, t )

 U ( x)( x, t )  i
2
2m x
t
2

(r, t )

 2 (r, t )  U ( x)(r, t )  i
2m
t
 2 d 2  ( x)

 U ( x) ( x)  E ( x)
2m dx 2
2
Q    * ( x, t )Qˆ  ( x, t )dx
pˆ  i
Q  (Q 2  Q )

x
Harmonic oscillator wavefunctions:
1
 b  2   1 2 b 2 x 2
0 ( x)  
 e
  
 b
2 ( x)  
8 
 b
1 ( x)  
2 
1
1
  1 b 2 x 2
 2
 (2bx)e  2 

 m 
b 2 
 
  1 2 b 2 x 2
 2
2 2
 (4b x  2)e

1
4
Gaussian integrals:

e

a ( z b) 2
dz 



 ze
a
a ( z b) 2
dz  b


b
2 bz

az
dz  e 4a
e
a


2

2  az
 z e dz 

2

a
1 
2 a3
 trans ktrans
2
R+T=1
E
particle flux
 k
2
T
 inc kinc
2
 r  ei ( kx t) ;  l  e i( kx
t)
2
k2
for free particle
2m
   (k )


 ( x, t )  ~ (k )e i ( kx  w( k )t ) dk v phase  f 
k

Lz  ml 
L   l (l  1)
J  L  S;
J 
S z  ms  (ms  s,s  1,....s  1, s)
v group 
d (k )
dk
S   s( s  1)
j ( j  1) j  l  s , l  s  1,..., l  s
J z  m j ; m j   j ,  j  1,... j
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Symmetric and antisymmetric spatial wavefunction:
S ( x1 , x2 )   n ( x1 ) n ( x2 )   n ( x1 ) n ( x2 )
A ( x1 , x2 )   n ( x1 ) n ( x2 )   n ( x1 ) n ( x2 )
Fermi-Dirac Distribution N ( E ) 
1
Be
 E / k BT
1
D( E ) 
dn
dE
Useful constants:
me  9.11  10 31 kg
m p  1.67  10 27 kg
1eV  1.6  10 19 J
h  6.63  10 34 J  s
k B  1.38  10 23 J / K
c  3.0  10 8 m / s
  1.05  10 34 J  s
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