PY354 Modern Physics
Final Exam
December, 16, 2003
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 120 points. Good luck!!
Problem 1. (20 pts)
Consider a square well with infinite sides of width L:
(a) (8 pts) Calculate, symbolically, the energy and wavelength of a photon emitted when a
transition between the n=5 and the ground state is made.
(b) (4 pts) Write down the expression for the probability that a particle in the nth state will be
found in the first 1/3 of a well of width L.
1
(c) (4 pts) Compare, with respect to unity, and as a function of the quantum number n, the
ratio of the probability that the particle will be found in the first third of the well to the
probability that it will be found in the middle third. Write down the expression and argue
mathematically, but do not solve the integrals. Feel free to make a graph.
(d) (4 pts) In quantum mechanics, we often talk about the correspondence principle. What is
the correspondence principle, and what does it mean?
Problem 2. (16 pts)
The energy levels of a 3D quantum box, of unequal lengths Lx, Ly, Lz on each side are given by:
2
 2 2  nx2 n y nz2 
Enx ,ny ,nz 

 .

2m  L2x L2y L2z 
2
(a.) (10 pts) If Ly  2 Lx  Lz write down the energy levels and degeneracy for the first 7 states.
3
Hint: First simplify the above equation, then count the energy of the states in units of
2
2
2mL2x
2
(b) (6 pts) Explain why the second x-direction state is not reached until the ?? energy level. A
conceptual explanation here is fine.
Problem 3. (26 pts) Schrödinger Equation: quantum mechanical tunneling
In this problem, you will derive the approximate transmission probability of a particle with
energy E incident on a potential barrier of energy U, U>E. In the figure below the wavefunction
in region I is shown split into two parts, one for the incoming wave and one for the reflected
wave. The number of particles per unit time which impinge on the barrier is given by
2
S   I  k I  where kI+ is the wavevector (related to the group velocity). The transmission
probability is given by the ratio of the outgoing flux to the incoming flux, or
 III k III
2
T
 I  kI 
2
(a) (8 pts) Identify the three regions in the figure and then write down the Schrödinger equation
and the trial wavefunctions for each of the three regions. Be sure to write down the value of the
wavevector in each region.
U
Energy
U=0
x=0
Position
x=L
3
(b) (10 pts) Write down the boundary conditions for this problem, and then apply each of the
boundary conditions appropriately to yield the set of coupled equations. Do not solve the coupled
equations.
E E
1  
U U
is the equation that would result from solving the
2m(U  E ) 
E E
L   4 1  
U U

4
T

sinh 2 

equations above.
(c) (8 pts) Make the following approximations to find the transmission probability in simpler
form: Assume that the height of the barrier is high relative to the incident energy E of the
particles, and that the barrier is relatively wide such that kII L >>1, where kII is the wavevector
(real) in the barrier region. Show, finally, that the transmission is proportional to ~ e 2 kL
4
Problem 4. (8 pts) Conceptual elements of quantum mechanical tunneling:
(a) (4 pts) Imagine that a barrier looks like
the figure to the right. Order the regions
in terms of the wavelength in each of the
three regions, from shortest to longest
wavelength. For this case E>UII.
E
U=0
UI
UII
UIII
(b) (4 pts) Calculate the ratio of the kinetic energy difference between region I and region III
if the former has a wavelength a factor of two different than the latter.
Problem 5. (12 pts) Addition of angular momentum
(a) (6 pts) Consider an atom which has a single electron in the n=3 state. Determine the possible
values of the total angular momentum.
5
(b) (6 pts) Now consider an atom which has two electrons in the d-shell. Determine the possible
values of the total angular momentum. Does the total symmetry of the wavefunction change with
j?
Problem 6. (18 pts) Hydrogen atom:
(a) (4 pts) In the hydrogen atom, for fixed orbital quantum number l, what happens to the
number of radial anti-nodes as the principle quantum number n goes up? Explain.
(b) (4 pts) In the hydrogen atom, for fixed principle quantum number n, what happens to the
number of radial anti-nodes as the orbital angular quantum number l goes up? Explain.
Feel free to draw.
6
(c) (10 pts) The radial probability density in the Hydrogen atom is given by
r
dP
2
P(r ) 
 r 2  R(r )  where the radial wavefunction takes the form Rn,n1 (r ) ~ r n1e na0
dr
for principle quantum number n. Find the most probable location of the radial component
of the wavefunction for any n.
Problem 7 (10 pts) Wave packets:
(a) (6 pts) The equation for the dispersion of a massive particle is:
for a massless photon is
velocities.

mc 2
k2

, and
2m
  ck . For each case determine the phase and group
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(b) (4 pts) Show how the group velocity is related to the particle’s classical velocity for
massive particles.
Problem 8 (10 pts): Molecules and Symmetrization
(a) (2 pts) What is the difference between Fermions and Bosons in terms of the symmetry of
the total wavefunction?
(b) (8 pts) For two electrons in a simple square well we argued that the ground state is
generally found to be a spin triplet. On the other hand, for the two electrons in a
Hydrogen molecule, the ground state is generally found to be a spin singlet. Explain this
difference, and show what the total wavefunction is in both cases.
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Formulas for PY354 Final Exam, Dec 16th, 2002
h
n 2 2 2
p   k
E  hf  
E
for infinite well
KEmax  hf  

2mL2
 2  2  ( x, t )
 ( x, t )
 2 d 2  ( x)


U
(
x
)

(
x
,
t
)

i


 U ( x) ( x)  E ( x)
2m x 2
t
2m dx 2
2 2
(r, t )

  (r, t )  U ( x)(r, t )  i
2m
t
2

pˆ  i
Q    * ( x, t )Qˆ  ( x, t )dx
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 2 b 2 x 2
 2
 (2bx)e

 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 


a

;
 ze

a ( z b) 2

b2
2 bz

az
dz  b
; e
;
dz  e 4a
a 
a



 trans k trans

z
2
e az dz 
2

1 
2 a3
2
R+T=1
E
 k
2
particle flux
T
 inc k inc
2
 r  ei ( kx t) ;  l  e i( kx t)
2
1 x x
k2
for free particle sinh( x )   e  e 
2
2m


 ( x, t )  ~ (k )e i ( kx  w( k ) t ) dk v phase  f 
k

   (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
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 )
Useful constants:
me  9.11  10 31 kg
m p  1.67  10 27 kg
c  3.0  10 8 m / s
1eV  1.6  10 19 J
h  6.63  10 34 J  s
  1.05  10 34 J  s k B  1.38  10 23 J / K
9