CHEM 342. Spring 2002. PS#3 Answers PS#1 Answers PS#2 Questions PS#2 Answers.doc (Word 97)
Problem Set #
2 Answers
Note:
Relevant Chapters in Mortimer are Chapters 14 and 15.Orthogonality
1. Prove that the functions 
and 
are orthogonal 
.
Hint: 
2. Give a mathematical definition for the Kronnecker delta 
. What is the numerical value of the Kronnecker delta when the two eigenfunctions are orthogonal? What is the numerical value of the Kronnecker delta when n and m are the same eigenfunction (i.e. n = m)? In addition to these two values, can the Kronnecker delta be equal to any other numerical values?
The eigenfunctions 
and 
are orthogonal when 
. The other possible value of the Kronnecker delta is 1, and it occurs when n = m. The Kronnecker delta can only equal zero or one; no other values are possible.
Operators
3. Find the result of operating with 
and 
on the function
. Is f(y) an eigenfunction of 
or of 
?
This function is not an eigenfunction of .
This function is an eigenfunction of with eigenvalue of zero.
4. Find the following commutators for any function f(x).
(a)
(b)
Hint: 
, 
, 
, and 
.
(a)
Therefore 
(b)
Therefore 
5. Find the result of operating with the operator 
on the function 
. What values must the constants have for 
to be an eigenfunction of 
?
For y
to be an eigenfunction of , the constant b must equal 1, while A can be any real number. In this case, the eigenvalue is 1.
6. Find the result of operating with the operator 
on the function 
. Is it an eigenfunction?
This is not an eigenfunction of .
7. The function 
is a well-behaved wave function in the interval 
. Calculate the normalization constant (A), and the average value of a series of measurements of x (i.e find the expectation value: 
).
Expectation Values
8. For the wave function and the operator
, give an expression that could be used to calculate the average value obtained from repeated measurements (i.e. show an expression for 
).
or
, where is normalized.
Particle In a Box
9. Calculate the value of A so that 
is normalized in the region 
.
Hint: 
Therefore, 
Note: 
for n = integer.
10. For a particle in a one-dimensional box 
, we used eigenfunctions of the form 
. Explain why we could not use
- The wave function must be zero for x = a. If k is a real number, then
cannot fit this boundary condition. - The boundary conditions also require that
when x = 0. This cannot be true for the cosine function because cos 0 = 1. Therefore, the allowed eigenfunction must be of the form
11. The ground-state wave function for a particle confined to a one-dimensional box of length L is 
. The box is 10.0 nm long. Calculate the probability that the particle is between 4.95 nm and 5.05 nm. Hint:
12. What is the ground state energy (i.e. n = 1) for an electron that is confined to a box which is 0.2 nm wide. [Hint: Planck's constant, h,is
J s; the mass of an electron, me, is 
kg]
Uncertainty
13. The speed of a certain proton is 4.5 ´
105 m/s along the x-axis. If the uncertainty in its momentum along the x-axis is 0.010 %, what is the maximum uncertainty in its location along the x-axis (i.e. 
)?
Tunneling
14. The wave function inside an infinitely long barrier of height V is 
. Calculate (a) the probability that the particle is inside the barrier; and (b) the average penetration depth of the particle into the barrier (i.e. the expectation value 
). Because the barrier is infinitely long, this wave function is valid for 
. Hint: 
