1
Physics 324 Problem Set # 6 Due 11/24/03
1. The Parity operator in one dimension, P1, reverses the sign of the position coordinate, x:P1Φ(x)=
Φ(−x). Note that P2
1Φ(x)=P1P1Φ(x)=Φ(−−x)=Φ(x)
(a) Find the eigenvalues of the Parity operator, P1. That is, find those values, λ, such that the
equation P1|Φi=λ|Φihas a solution.
(b) Find the eigenfunctions of P1.
(c) Is the Parity operator, P1, Hermitian? Why, or why not. Remember, A†is the Hermitian Adjoint
of an operator Aif and only if hA†f|gi=hf|Agifor all vectors |fiand |gi.
(d) Derive an expression for [P1,ˆx], the commutator of the parity operator with the position operator.
(Hint, make use of a trial vector.)
2. Problem 3.40 in your textbook.
3. For operators, ˆ
Q, that do not depend expliciltly upon time, we have:
d
dt hˆ
Qi=i
¯hhˆ
H, ˆ
Qi
Using the results from Problem 3.41 in your text,
(a) For ˆ
H=ˆp2/2m+V(x), compute dhˆxi/dt.
(b) For ˆ
H=ˆp2/2m+V(x), compute dhˆpi/dt.
(c) For the harmonic oscillator Hamiltonian (Eqn. 2.39), compute dhˆ
P1i/dt, where P1is the parity
operator from Problem 1.
4. Problem 3.48 in your textbook.
5. Problem 3.57, parts (a) and (c), in your textbook.