1
PHYS234 Elementary Quantum Mechanics I
Sample Final Examination Questions


Formulae:
NOTE: not every formula listed here is needed in answering this exam paper

1D time-independent Schrödinger equation

2 2
2
( ) ( ) ( )
2
d
V x x E x
m dx
 
 
   
 

(1)
Time-dependence of stationary state

/( , ) ( )iEtx t e x   (2)
First quantization rule: xˆ x , ˆ
d
p i
dx
  (3)
Eigenstates of momentum operator pˆ (plane waves): /
1
2
ipxp e

 

(4)

Eigenstates of position operator xˆ : ' ( ')x x x  (5)

Commutation relation [ , ] [ , ] [ , ]A BC B A C A B C  (6)

Time evolution of expectation value
 ,
d A i
A H A
dt t

 
 
(7)

3D time-independent Schrödinger equation with a central potential:

2
2 ( )
2
V r E
m
     

(8)
Laplacian operator in Cartesian and spherical coordinates:

2 2 2 2
2 2
2 2 2 2 2 2
ˆ1 L
r
x y z r r r r
     
      
      
(9)
Solution of the hydrogen atom:
( ) ( ) ( , )
s s
m
s nlm m nl l mnlmm R r Y     r (10)
First few normalized functions are:

/
10 3/ 2
/ 2
20 3/ 2
/ 2
21 3/ 2
2
1 1
1
22
1
24
r a
r a
r a
R e
a
r
R e
aa
r
R e
aa




 
  
 


1/ 2
0
0
1/ 2
0
1
1/ 2
1
1
1
4
3
cos
4
3
sin
8
i
Y
Y
Y e 





 
 
  
 
 
  
 
 
  
 

(11)
where 2 2
04 /a me  is the Bohr radius.

Energy levels of a hydrogen atom:

2

2
2 2 2
1 13.6eV
2
nE
ma n n
   

(12)

Normalization of radial and angular parts
2 2
0
( ) 1R r r dr

 and
2 2
0 0
sin 1mlY d d
 
     (13)

Orbital angular momentum operators in spherical coordinates:

2
2 2
2 2
1 1ˆ sin
sin sin
ˆ
z
L
L i

    

    
     
    

 



(14)
Commutation relations of the angular momentum operators:

2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ[ , ] , [ , ] , [ , ]
ˆ ˆ[ , ] 0 where , ,
x y z y z x z x yL L i L L L i L L L i L
L L x y z 
  
 
  
(15)
Eigenstates and eigenvalues of 2Lˆ and ˆzL :
2 2ˆ ˆ( 1) , .m m m ml l z l lL Y l l Y L Y m Y    (16)
Coupling of two spin s = 1/2 particles, sSm :

1
2
1
2
00 ( ) singlet
11
1 1 triplet
10 ( )
   
 

   
    
(17)


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Question 1
For a general quantum mechanical problem with Hamiltonian H, we usually denote its
normalized stationary states as ...abc , where a, b, c, … are good quantum numbers
corresponding to eigenvalues of operators A, B, C, … respectively.
a) State the properties of the operators A, B, C, …. No mathematical expressions are
necessary.
b) Without knowing the mathematical form of the wavefunctions ...abc , argue that
they must be orthogonal, i.e., a a b b c ca b c abc           .

An electron of a hydrogen atom is in the superposition state
 
/
/2 1 1 0
1 1 13/2
2
7
4
r a
r ae rA e iY Y Y
a a
  


 
 
 
     
 
, (18)
where a is the Bohr radius, A is a normalization constant, and mlY are spherical
harmonics.
c) First express  as a linear combination of hydrogen atom solution snlmm as
defined in Eq.(10). Then use the orthogonality relation of snlmm to show that
1
12 6
A  , and that
1 1 1 1
2 2 2 2
1 1
100 211 21 1 7 210
2 3 2
i          
 
(19)

d) If you measure the energy, what are the possible outcomes and corresponding
probabilities?
e) One of your classmates claims that since  is a combination of different energy
levels, the electron will spend part of its time in those levels and therefore its
energy is not conserved. Suppose the only way to know the actual energy of the
system is to perform an experiment to measure it, give a simple qualitative
argument (no calculation) to convince your classmate that the behaviour of this
system does not violate the conservation of energy.
f) If you measure Lz, what is the probability of getting zero?
g) What are the possible outcomes if you measure 2J ?
h) What are the possible outcomes and probabilities if you measure
zJ ?
i) Explain why the radial distance r and zS can be measured simultaneously. If you
measure r and zS , what is the probability density for finding the electron with
spin down ( / 2zS   ) and radial distance r?









continue on next page …

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Question 2
Consider a particle of mass  (do not use m, otherwise will confuse with a quantum
number in this problem) moving in the xy plane under a two-dimensional central
potential ( )V r . The coordinates of the particle is expressed as ( , )r  as shown.

The Hamiltonian of the particle is
2 2 2
2 2 2
1 1
( )
2
H V r
r r r r 
   
     
   

.
a) Show that the wavefunction ( , )r  can be written as ( , ) ( ) imr R r e    , where
( )R r is independent of  .
b) Explain why m cannot be an imaginery number.
c) In the above notation, m can be chosen to be non-negative. Explain what values of
m are allowed.
d) The two-dimensional angular momentum operator is defined as L i     . Do
H and L have simultaneous eigenfunctions? If yes, what are they? If no, explain
why.

Suppose the particle is in a two-dimensional infinite circular well,

0,
( )
,
r a
V r
r a

 
 
. (20)
e) Show that the radial equation is

2 2 2
2 2
1
2
d d m
R ER
dr r dr r
  
     
  

. (21)

You are given that eigenvalues and eigenfunctions of the radial equation (21) are
2 2
22
nm
nm
x
E
a


and ( )nmR r , 0,1,2,3,...n  . The values of nmx are shown below:
m n = 0 n = 1 n = 2
0 2.4048 5.5201 8.6537
1 3.8317 7.0156 10.1735
2 5.1356 8.4172
3 6.3802 9.7610

f) Show that each energy level
nmE is two-fold degenerate except when m = 0. Note
that you have to find two orthogonal wavefunctions having the same energy.
What is the degeneracy of the m = 0 energy level?
g) We use the notation s, p, d, f, … to denote states with angular momentum
quantum number m = 0, 1, 2, 3, … respectively. What are the five lowest energy
states in this notation (for example, a state with n = 0 and m = 0 should be denoted
as 0s )? List them in increasing order of energy.


continue on next page …
r

x
y

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h) Suppose we put two identical non-interacting spinless (S = 0) particles in this
circular well, what are the energies and normalized total wavefunctions
1 1 2 2( , , , )r r  in the ground state (lowest energy) and first excited state (next
lowest energy) respectively. You should express the total wavefunctions in terms
of
nmR .
i) Repeat part h) if the two particles are electrons, again assuming that they are non-
interacting, i.e., ignore electron-electron repulsion. Remember that since the spin
is nonzero, wavefunctions have both orbit and spin parts.
j) Discuss how electron-electron repulsion splits the first excited state in part i).
Give a qualitative reason why one has higher energy than the other. No
mathematical expression or proof is necessary.

Now consider a particle in a three-dimensional infinite cylindrical well, i.e., it is
inside a cylinder which is infinitely long along the z direction and has radius a in the
x-y plane.
k) Find its energies and wavefunctions ( , , )r z  using your knowledge of the two-
dimensional infinite circular well above. Give conditions and allowed values for
any variable or quantum number you introduce. Are the energy levels discrete or
continuous? Are the wavefunctions normalizable?


Question 3
In general, the Hamiltonian of a one-dimensional particle is H T V  where
2
2
p
T
m
 is the kinetic energy operator, ( )V x is the potential.
a) From the time evolution of expectation values, Eq. (7), show that
2
d dV
xp T x
dt dx
  . (22)
b) Show that for stationary states,
1
2
dV
T x
dx
 .
Consider a one-dimensional “hydrogen atom”, i.e., a particle in one-dimension in a
potential ( )V x
x

  , where 0  is a constant.
c) Note that the potential is an even function, ( ) ( )V x V x  . State its implication on
the stationary states. No proof is necessary.
d) In general, what is the lower bound of the energy of a quantum mechanical
particle in a potential V? Does this lower bound exist in this “hydrogen atom”?
e) Suppose ( )x is the ground state wavefunction, i.e., the lowest energy state.
Explain why (0) should not be zero.
f) Using the result in part b), find the energy of this ground state. Does it contradict
part d)?
g) Judging from T of the ground state, guess what the wavefunction must be. Give
your reason, but no need to give a mathematical proof.
h) Give an example of a one-dimensional potential whose minimum is  but
whose ground state energy is finite. No proof is necessary.












































































































































































































































































































































































































































































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