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Physics 943 - Homework 4
Department of Physics Due February 27, 2009
University of New Hampshire
1 Gaussian Wave
Consider a one-dimensional quantum particle with a gaussian wave function
Ψ(x) = C exp(a x
2
where a, b and C are complex numbers. We will limit ourselves to the case Re a < 0.
(a) Calculate the norm
dx|Ψ(x)|
2
of this wave function.
(b) Calculate the momentum-space wave function
Ψ(p) and show that it also has
a gaussian form
Ψ(p) =
C exp(˜a p
2
b p). (2)
(c) Verify that the momentum-space and the coordinate-space wave functions
have the same norm.
(d) Calculate the expectation values 〈x〉, 〈p〉 and the uncertainties ∆x and ∆p.
Show that for any gaussian wave function, ∆x · ∆p ≥ ¯h/2 and that the
equality is achieved whenever a is real.
2 Time Evolution
(a) Prove that for any operator
B for which
[
B,
H
]
= 0 we have
B
〈α, t|
B |α, t〉 = 〈α, t = 0|
B |α, t = 0〉, for any state |α, t〉 at any time t, in
other words, the observable
B is a constant of the motion.
(b) (Sakurai 2.4) Let xˆ(t) be the operator for a free particle in one dimension in
the Heisenberg picture. Evaluate [ˆx(t), ˆx(0)].
(c) (Sakurai 2.5) Consider a particle in one dimension whose Hamiltonian is
given by
H =
2 m
ˆp
2
By calculating
[[
H, xˆ
]
, ˆx
]
, prove
a
′
|〈a
′′
|ˆx| a
′
〉|
2
(E a
′ − E
a
¯h
2
2 m
where |a
′ 〉 and |a
′′ 〉 are energy eigen kets with eigenvalue E a
′ and E a
′′ respec-
tively.
(d) (Sakurai 2.6) Now consider the same Hamiltonian in three dimensions:
H =
2 m
ˆp
2
by calculating
[
ˆx · ˆp,
H
]
obtain:
d
dt
〈ˆx · ˆp〉 =
p
2
m
− 〈ˆx · ∇V 〉 (6)
For any stationary state we have 〈xˆ · ˆp〉 independent of time, so the left hand
side of this equation is zero and we have the Quantum Mechanical equivalent
of the Virial Theorem.
3 A Spin Rotator
In this problem we explore the consequences of making a sequence of infinitesimally
different measurements of the spin of a particle by a set of Stern-Gerlack like
apparati. For notational simplicity we set μ 0
= 1 in this problem. Recall from
HW2 that we worked out the formalism for measuring the spin of a silver atom
in an arbitrary direction ~n, with |~n|
2
= 1. Another way of forcing the norm of
~n to be 1 is to choose it’s components as ~n = (sin θ cos φ, sin θ sin φ, cos θ). The
normalized eigen kets |~n, ±〉 of the 2 × 2 matrix ~σ · ~n where the ~σ are the usual
dθ
M ( θ)
M ( dθ)
x
z
θ
Figure 1: The angles of the many Stern-Gerlach apparati.
(ii) Find the coefficient g(θ) that determines the norm of the state produced
by this sequence of measurements. (Hint: you can derive a differential
equation for g(θ) by working out what g(θ + dθ) |θ + dθ〉 is, similar as
our work in lecture for translation in time or space.)
(iii) What do yo conclude?
