Physics 410 Quantum Mechanics Fall 2015
Homework #4 Sept 23, 2015 due Wednesday, Oct 7, 2015
Griffiths problems 2.11 (a)-(b), 2.12, 2.16
Johnson 4.1 In Example 2.4, Griffiths shows how to use the raising operator a
+
to get the first excited harmonic oscillator state, state wfn ψ
ψ
1
(x) from the ground
0
(x). Apply again to get the second excited state, ψ
2
(x) (do not worry about overall normalization). Does this agree with what we find in section 2.3.2? All wavefunction here are in coordinate space!
Johnson 4.2 Consider the wavefunction 𝜓 𝑥 =   𝐴   𝑥 exp   − 𝑎 + 𝑖𝑏 𝑥 , 𝑥   ≥   0 , 𝜓 𝑥 =   0 , 𝑥   <   0 .
Assume a and b are real positive numbers.
(a) What are the dimensions of a, b, and A ?
(b) Find the normalization constant A . Do you get the correct dimensions?
(must answer the question for full credit).
(c) Sketch the wavefunction and indicate roughly where you think < x > should be.
(c) Compute < x >. Does it meet your expectation from (c)? Does it have the right dimensions?
(d) Compute < p >.