Download Quantum Theory 1 - Class Exercise 5

Quantum Theory 1 - Class Exercise 5
1. Particle in a constant potential.
Consider a particle of mass m that is moving in a one dimentional potential V (x).
Assume that in a certain region of space the potential is a constant i.e. V (x) = V . In
that region, find a general solution for the time independent Schroedinger equation for
the conditions
(a) E > V .
(b) E < V .
(c) E = V .
where E is the particles energy.
2. Wave packet probability current density.
Show that for any square integrable wavepacket the relation
Z ∞
hpi(t)
J(x, t)dx =
m
−∞
(1)
holds. Where J(x, t) is the probability current density, and hpi is the mean momentum
of the particle.
3. Free particle probability current density.
Consider the wave packet
h
i p2
p
p
ψ(x, t) = Aei ~ x + Be−i ~ x ei 2m~ t .
(2)
Calculate the probability current density.
Show that it is zero for a particle in any eigenstate of an infinite well.
4. Particle on a ring.
Consider a particle of mass m that is moving in a one dimension on a ring of circumference L. The state of the particle is given by
2π
4π
2π
1
A
1
ψ(x, t) = √ ei( L x−ω0 t) + √ ei( L x−4ω0 t) + √ ei(− L x−ω0 t)
3L
6L
L
1
(3)
(a) Find ω0 .
(b) Find |A|.
(c) What are the possible values of momentum which can be measured in the system
and with what probability?
(d) What are the possible values of energy which can be measured in the system and
with what probability?
(e) Calculate the expectation values for the momentum and the energy of the particle.
In a measurement of the particle’s energy we measured E =
2~2 π 2
,
mL2
after
the measurement
(f) What would be the state of the particle?
(g) What are the possible values of momentum which can be measured in the system
and with what probability?
where E is the particles energy.
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