Download III. Consider three non-interacting particles of masses M, 2M and 4M

Chemistry 4521
Problem Set 11 Solutions
Note: Solutions to Problems I and II are contained in the Friday lecture notes, and are already
posted.
Consider three non-interacting particles of masses M, 2M and 4M,
constrained to lie in a square with sides of length L. How many quantum
numbers are there for this system? Separate variables and obtain the
eigenfunctions and eigenvalues. What are the degeneracies of the three
lowest energy levels?
III.
There will be 6 quantum numbers.
Call the three particles 1,2,3 respectively.
Incorporate the relative mass into the quantum number part of the
expression.
 22 
2
2
nx22 n y 2 nx23 n y 3 



En x , n y , n z 
n  n 
 n x1  1, 2, , n y1  1, 2, , n x2  1, 2,
2 ML2 
2
2
4
4 
The wave function is a product of six particle in a box eigenfunctions, as you
know.
2
x1
2
y1
 2 2
Let us express all energies in units of
. We make up a list quantum
2ML2
numbers and energies for all states n = 3 or less. We then arrange them in
order of increasing energy.
Results:
The lowest level is (1,1,1,1,1,1,) with energy 3.5. It is non degenerate (or
singly degenerate, if you prefer).
Second is (1,1,1,1,2,1) with energy 4.25. It is twofold degenerate. The
(1,1,1,1,1,2) state has the same energy.
1
Third is (1,1,1,1,2,2) with energy 5. Other levels with the same energy are
(1,1,2,1,1,1) and (1,1,1,2,1,1).
Thus the third energy level is three-fold degenerate.
**********************
This may be the end of the problem, but we need to check.
Going higher in energy,
(111113) and (111131) are at energy 5.5
and ((111123) and (111132 are at energy 6.25
So it looks like all the remaining states similarly have energies > 5, and we
are done.
IV. Find the wavefunctions and energy levels for a particle of mass M
contained in a cube with sides of length L. What is the degeneracy of the
level with energy three times greater than that of the ground state?
En
x
n
,ny ,nz
x

2
n
2ML
2
2
x
 ny2  nz2
3
2
,ny ,nz

nx  1,2,, ny  1,2,, nz  1,2,
n x
 
 x , y , z    2L  sin  x L
 

 ny  y

sin




 L

 n z 
 sin  z
 inside the cube, and zero outside

L



The ground state (1,1,1) has energy 3 in units of
2
2ML2
. Thus to answer the
question, we wish to find which levels have energy 9?
The answer is (2,2,1), (2,1,2) and (1,2,2). Any quantum number >2 will
exceed the value 9 for the total energy. The level is said to be threefold
degenerate
2