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CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA
CHEMISTRY 312 - HOMEWORK #2
The Problems below come from McQuarrie, Quantum Chemistry. They correspond to the
following problems in the text:
Chapter 3: 4, 6, 12, 13,and 22
1. Show that ( )( )( )cos ax cos by cos cz is an eigenfunction of the operator: 2 2 2
2
2 2 2V
x y z
! ! != + +! ! !
Which is called the Laplacian operator.
2. Determine weather or not the following pairs of operators commute:
2
2
ˆ ˆ
( ) 2
( )
( )
( )
A B
d d dadx dx dx
db x
dx
c SQR SQRT
dx y
+
! !
! !
3. A discrete probability distribution that is commonly used in statistics is the Poisson distribution:
0,1,1,!
n
nf e nn
!! "= = …
4. An important continuous distribution is the ex-ponential distribution
( ) 0xp x dx ce dx x!
= " < #
Evaluate 2, , , and the probablity that x .c x ! "#
5. Using the trigonometric identity ( ) ( )1 1
sin sin = cos cos2 2
! ! " !# # $ $ +
Show that the particle-in-a-box wave functions (Eq. 3-28) satisfy the relation:
( ) ( )*0
a
n mo
x x dx m n! ! = "#
(The asterick in this case is superflous because the functions are real.) If a set of functions
satisfies the above integral conditions, we say that the set is orthogonal and, in particular,
that ( )mx! is orthogonal to ( )
nx! .If, in addition, the functions are normalized, we say that
the set is orthonormal.
Additional Problems:
One of the key theorems in quantum mechanics is that it is possible to expand any exact
wavefunction ! of a form of Schroedinger’s Equation that has no analytical solution in terms of
the set of wavefunctions !i of a known system as long as both systems satisfy the same boundary
conditions. This is because the functions !i form a complete set over the spatial region in which
they are defined. This problem illustrates this theorem.
The exact wavefunction can be written as a linear sum of the !i:
!
" = aii=1
n
# $i,
where
a
i= ! *
" # d$ ,
if both !i and " are normalizable wavefunctions. Consider the !i to be the usual wavefunctions
for the particle-in-a-box potential with infinite potential walls at 0 and L.
Now, somehow, we will prepare a system in which the true wavefunction is
"(x) = Nx[x(x-L)]
Where Nx is a normalization constant and "(x) is a parabolic wavefunction in the potential well.
Using the above equations, determine the coefficient ai for expanding "(x) in terms of the !i for
values of i up to 6 (the necessary integrals can be found in the CRC or other compilations).
Now Plot
!
"(x) and aii=1
n
# $i(x)
from 0 to L for n = 1 to 6 and compare the various approximations to the “actual” wavefunction.
You may write a program to generate data files containing the various approximations, or use a
program like Igor or a spreadsheet to do the calculations and plots.