Problem Set #2 Answers
Note:Relevant Chapters in Mortimer are Chapters 14 and 15.
1. Prove that the functions and are orthogonal .
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.
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).
Hint: , , , and .
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: ).
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 ).
, where is normalized.
Particle In a Box
9. Calculate the value of A so that is normalized in the region .
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
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]
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. )?
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: