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

    1. The wave function must be zero for x = a. If k is a real number, then cannot fit this boundary condition.
    2. 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: