Final exam
Physics 206, 2007
1. (6 points total) Pauli matrices.
0 1
 is a Pauli matrix.
a) (3 points) Calculate the matrix sin(  x / 6) , where  x  
1
0


b) (3 points) Calculate the matrix cos( [ y   x ] / 2 ) , where the Pauli matrix
0  i
.
0 
 y  
i
2. (4 points total) Trapped particle. A particle of mass m is confined in one dimension
and moves in the potential U 0 ( x) which is given by the following equations in three
regions | x | a; a | x | 2a and 2a | x | : U 0 ( x)  0 at | x | a and | x | 2a ;
U 0 ( x)  V0  (2a  | x |) at a | x | 2a . The particle is initially localized in the region
| x | a and its energy E  aV0 . In the limit of large V0  0 , estimate the typical time t 0
before the particle leaves the region | x | a .
3. (10 points total) Resonant scattering. A particle of mass m with the low energy
2k 2
moves in a spherically symmetric potential U (r ) . The potential U (r )  0 at
E
2m
r  a .There is a bound state  (r ) in the potential U (r ) . We assume that the energy of
 2 2
is close to zero. Find the energy dependence of the total
2m
scattering cross-section. You can assume that only s-scattering is relevant.
this bound state   
a) (2 points) Find  (r ) at r  a .
b) (1 point) Calculate
1 d [r (r )]
at r  a .
r (r )
dr
c) (2 points) At r  a , express the spherically symmetric solution of the Schroedinger
equation with the energy E via the scattering phase  0 .
d) (2 points) Use the result of problem 3b) to find cot  0 . Assume that ka   0 .
e) (3 points) Calculate the scattering cross-section.
exp( 2i 0 )  1
[Hint: the scattering amplitude f 
]
2ik
4. (15 points total) Two-dimensional atom. Electrons are confined in two dimensions
and move in the potential U ( x, y)  m 2 ( x 2  y 2 ) / 2 . Due to the screening of the
electrostatic interaction, the inter-electron interaction can be represented in the form
 
 


V (r1 , r2 )  V0 (r1  r2 ) , where r1 and r2 are the positions of the electrons.
a) (2 points) Find the degeneracy of all energy levels of the two-dimensional atom with
one electron. Don’t forget the electron spin!
b) (1 point) Write down the ground state energy and all linearly independent ground state
wave functions of the one-electron two-dimensional atom.
[Hint: the ground state wave function of the one-dimensional oscillator has the form
 ( x)  A exp( mx 2 /[ 2]) . ]
c) (1 point) The same question for the first excited energy level of the one-electron atom.
[Hint: the wave function of the first excited state of the one-dimensional oscillator has the
form  ( x)  Bx exp( mx 2 /[ 2]) . ]
d) (1 point) Neglecting the electron interaction, write down all linearly independent
ground state wave functions for the two-electron atom.
e) (2 points) The same question for the first excited level of the two-electron atom. (How
many linearly independent states are there in this level?) To make equations shorter,
simply write two-particle wave-functions as combinations of single particle wavefunctions.
f) (3 points) Assume that V 0 is small and find the correction to the ground state energy
of the two-electron atom in the first order perturbation theory.
g) (5 points) The same question for all states in the first excited level. [Hint: these states
are degenerate at V 0 =0 but they do not have to be degenerate at V0  0 .]