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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.