This preview shows page 1. Sign up to view the full content.
Unformatted text preview: PHY4221: Homework 1 Due Sep. 11, 2008 1. A particle of mass m is conﬁned in a three-dimensional potential with the ground state
− ψ (x, y, z ) = Ae y2
2 − 2σ 2
y − z2
2σz where σi (i = x, y, z ) are the widths.
(a) Find the probability density of the particle at time t if the potential is switched oﬀ
suddenly at t = 0.
(b) If σx = σy σz , discuss how shape of the probability density changes with time. (c) Consider a gas of classical particles at a temperature T , and it is conﬁned in a box with
the lengths σx = σy σz . If the walls of box suddenly disappear, the gas is released freely. Discuss how the shape of density of the gas compare with that in (b) as time increases.
2. The initial wave function of a free particle of mass m is given by,
ψ (x, 0) = Ae−α|x|
where A is the normalization constant, and α > 0.
(a) Find the normalization constant A.
(b) Find the momentum amplitude of ψ (x, 0) and hence the average energy of the particle.
(c) Estimate the width of the wave function at time t > 0.
3. An initial wave function of a free particle (mass m) is given by
x2 ψ (x, 0) = Ae− 2σ2 eiαx 2 which is a Gaussian wave packet with an extra phase factor in the last term, assuming real
α > 0. Discuss how the extra phase factor aﬀects the motion of the wave packet. Suggest a
physical way to add the phase factor to the particle wave function.
4. The momentum amplitude of an arbitrary wave function of a free 1D particle is deﬁned
by ϕ(k ) = √1
2π e−ikx ψ (x, 0)dx. Derive a relation of momentum density |ϕ(k )|2 and position |ψ (x, t)|2 in the long time limit. Explain the relation in terms of classical physics. 1 ...
View Full Document