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. 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
1
A
ψ(x, t) = √ ei( L x−ω0 t) + √ ei( L x−4ω0 t) + √ ei(− L x−ω0 t)
3L
6L
L
(1)
(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 =
the measurement
(f) What would be the state of the particle?
1
2~2 π 2
,
mL2
after
(g) What are the possible values of momentum which can be measured in the system
and with what probability?
3. Consider a Hamiltonian which describes a one dimensional system of two particles of
masses m1 and m2 moving in a potential that depends only on the distance between
them.
Ĥ =
p21
p2
+ 2 + V (x1 − x2 )
2m1 2m2
(a) Write down Schroedingers equation using the new variables
X=
m1 x1 + m2 x2
, x = x1 − x 2
m1 + m2
Interpret the meaning of these variables.
(b) Use seperation of variables to find the equations of motion of X and x.
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