CM2008 Quantum Mechanics Section Coursework
Hand solution to the following problems into Room 107 by 5.00pm on Monday
30 th April.
1) Which quantum mechanical problem is used to model the translational motion of a molecule? Give an expression for the energy levels of the molecule and define all the terms in the expression. What is the boundary condition of the problem? 5 marks
2) What three conditions does the Born interpretation place on the wavefunction and why? Where does the fourth condition come from? 5 marks
3) Write down the Schrödinger Equation for the vibrational motion of a diatomic molecule (atom 1 of mass m
1
, coordinate x
1
; atom 2 of mass m
2
, coordinate x
2
) defining all the terms in the equation. Give the expression for the quantized energy levels of the problem. 5 marks
4) Calculate the normalization factor, N , of the hydrogen 2p
1
atomic orbital.

2 , 1 , 1
( r ,

,

)

N ( r / a
0
) e  r / 2 a
0 sin
 e i

(information below)
5)
The Schrödinger Equation for rotational motion in 2 dimensions is:
 
2 mr
2
2

2

 
 
2

E
   and the wavefunctions are:
 
( 1 / 2

) 1 / 2 exp
 im
 
, m

0 ,

1 ,

2 ...
m
Calculate the energy of the m = 3 level and comment on its degeneracy. 5 marks
6) What are the values of the quantum numbers n , l and m of the hydrogen wavefunction, stating clearly how you identified them:
ψ
( r , θ , φ ) = N [ r
2
/ a
0
2 ] exp(r /3 a
0
) cos θ sin θ exp(i φ )
What Cartesian orbitals are obtained if this wavefunction is combined with the similar wavefunction
ψ
( r , θ , φ ) = N [ r
2
/ a
0
2 ] exp(r /3 a
0
) cos θ sin θ exp( i φ )?
5 marks
Information for Q4


1 , 0 , 0
( r ,

,

) 2 d
    

1 , 0 , 0
( r ,

,

) *

1 , 0 , 0
( r ,

,

) r 2 dr
0
 r
 
, 0
   
, 0
  
2
 sin
 d
 d
 standard integral :


0 x n exp(
 ax ) dx
 n !
/ a n

1