Homework 3
(due Friday, September, 13)
1. Prove the following commutator identities starting from the basic definition of the
commutator, [A, B] = AB − BA. Assume that the operators are linear and specifically
obey the property Acψ = cAψ for c any complex number.
[A, (c1 B + c2 C)] = c1 [A, B] + c2 [A, C]
[A, B] = − [B, A]
[A, BC] = [A, B] C + B [A, C]
(1)
(2)
(3)
2. Use the above identities to commute the following commutators.
(a) [p, x2 ]
(b) [p, x3 ]
(c) [x, p2 ]
(d) [x, p3 ]
(e) [x2 , p2 ]
3. In a manner similar to what was done in lecture prove that the if ψ is a wave function
for the harmonic oscillator with energy E, then a− ψ has energy E − h̄ω.
4. For the harmonic oscillator state
pectation values.
q
q
2/3 ψ2 + eiπ/4 1/3 ψ3 compute the following ex-
(a) Start by expressing x and p in terms of the raising and lowering operators for the
harmonic oscillator.
(b) Compute the expectation value of x.
(c) Compute the expectation value of p.
(d) Compute the expectation value of x2 .
(e) Compute the expectation value of p2 .
(f) Check the uncertainty principle.