PHY4604–Introduction to Quantum Mechanics
Fall 2004
Problem Set 7
Oct. 18, 2004
Due: Oct. 25, 2004
Reading: Notes, Griffiths Chapter 3
1. Maple problem: integrating Schrödinger’s equation. Download separately.
2. Drill.
(a) Linearity. Given the following operators, which are linear? Why?
i. ψ → x3 ψ
d
ii. ψ → x dx
ψ
3
iii. ψ → ψ
(b) Commutators. Calculate the commutators (L± are the SHO ladder operators defined in class):
d
i. [x3 , x dx
]
ii. [∇2 , 1r ]
d
iii. [x2 , dy
]
iv. [3, x2 ]
d
]
v. [3, dx
(c) Complex numbers. Simplify:
i. |3 + 2i|2
√
ii. Re x2 − 2ixy
iii. Im eiωt/h̄
(d) Projecting out amplitudes. Determine the amplitude desired in each
problem:
i. Expand the sawtooth function
f (x) =


 x + a/2
x
0


< −a/2 < x < 0
0 < x < a/2
otherwise
in terms of eigenstates of the infinite square well between (−a/2, a/2).
Determine the amplitude of the 1st and 2nd excited states.
1
(e) Delta functions. Evaluate:
i.
ii.
R3
−3
R3
−3
1 2
dx δ(x − 2) exp− 2 x
1 2
dx δ(x − 4) exp− 2 x
(f) Hermitian operators. Let x and p be the usual operators for position
and momentum. State whether each of the following operators is selfadjoint (Hermitian), anti-self-adjoint (antiHermitian, O† = −O), unitary
(O† O = 1), or if none of the above, what the adjoint is. Hint: remember
(AB)† = B † A† .
i. xxp
ii. xpx
iii. xpp + ppx
d2
iv. dx
2
(g) More operators. Two operators A and B satisfy A = B † B + 3 and
A = BB † + 1.
i.
ii.
iii.
iv.
show A is self-adjoint
find [B † , B]
find [A, B]
Suppose ψ is an eigenfunction of A with eigenvalue α. Show that if
Bψ 6= 0, then Bψ is an eigenfunction of A, and find the eigenvalue.
2