Homework Assignment 04
Continuous Spectra, Translation and Momentum, Canonical Commutators
(Due: 02/18/2016)
Problem 1 (40%) Consider a particle of mass m in the n-th stationary states of a one-dimensional
infinite potential well with width of l , whose wavefunction in position representation is
⎛ nπ ⎞
2
ψn (x) ≡ x n =
sin ⎜⎜⎜ x ⎟⎟⎟ .
l
⎝ l ⎟⎠
(a) [15%] Find the wavefunction in momentum representation ψn (p) ≡ p n .
(b) [25%] Evaluate Δx ⋅ Δp for this state, where Δx =
(ΔX )2 and Δp =
(ΔP)2 . Then
1
verify that the uncertainty principle, Δx ⋅ Δp ≥ ! , is held for this state.
2
[Useful integrals:
⎤
1 ⎡⎢ 2
1
2
⎥,
x
sin
(x)
dx
=
x
−
x
sin(2x)−
cos(2x)
∫
⎥
4 ⎢⎣
2
⎦
3
2
⎛
⎞
x
x
1⎟
x cos(2x)
2
2
∫ x sin (x)dx = 6 − ⎜⎜⎜⎜⎝ 4 − 8 ⎟⎟⎟⎠ sin(2x)− 4 . ]
Problem 2 (36%) For a Hamiltonian given by
P2
H=
+V(X )
2m
(a) [10%] Evaluate the commutators [H, X ] and [[H, X ], X ] . [Note: please use the fundamental
commutator and the properties of commutators to evaluate these. You need to master this
important skill.]
(b) [26%] Then prove that
2
2
(E
−
E
)
k
X
n
=
,
∑
k
n
2m
k
where En and Ek are eigenvalues of H , while n and k are the corresponding
eigenstates, respectively.
Problem 3 (24%)
(a) [6%] Let x and p be the coordinate and momentum in one dimension. Evaluate the classical
Poisson bracket {x,F(p)} , with F(p) a function of p.
PB
(b) [6%] Let X and P be the corresponding quantum operators, evaluate the commutator
⎡
⎛
⎞⎤
⎢X, exp ⎜⎜ iPa ⎟⎟⎥ ,
⎢
⎜⎝  ⎟⎟⎠⎥
⎢⎣
⎥⎦
where a is a constant.
(c) [12%] Prove that
⎛ iPa ⎞⎟
⎟⎟ x
α ≡ exp ⎜⎜⎜
⎝ ! ⎟⎠
is an eigenstate of operator X. What is the corresponding eigenvalue?
[Notes: you might find that the result in part (b) is useful. In this problem, we use the
fundamental commutator and momentum operator P to show that exp(iPa / ) is the space
translation operator; while in class we used the property of space translation operator to show
that P is the momentum operator.]