Download Quantum Theory 1 - Class Exercise 4

Quantum Theory 1 - Class Exercise 4
1. Consider a ”quantum dice”,which is just a quantum description of a regular dice.
We define the number operator N̂ ϕn = nϕn and the evenness operator Ẑϕn =
1+(−1)n
ϕn .
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(a) When measuring the number operator, what is the probability of getting N̂ = 4?
(b) When measuring the evenness operator, what is the probability of getting Ẑ = 1?
(c) We measure the number operator and get N̂ = 4.Afterwards, we measure the
evenness, what is the value we are going to get?
(d) We measure the evenness operator and get Ẑ = 1. Afterwards, we measure the
number operator. What is the probability of measuring n = 4?
2. Consider a free particle in an infinite well of length L. Find stationary states, eigenenergies, and eigenvalues of the momentum operator.
3. Consider a free particle in an infinite well of length L. At time t = 0 the particle is
prepared in the state
√
√
√
ψ(x, 0) = A ϕ1 + 2ϕ2 + 3ϕ3 + 2ϕ4 + ϕ5
(a) Find |A|
(b) Find ψ(x, t)
(c) What is the probability of measuring an energy larger then
(d) What is the probability of measuring a momentum of
4π~
?
L
(e) In a measurement of the particles energy we got the value
surement, we measured the particles momentum.
• What is the probability of having p =
2π~
L
• What is the probability of having p =
4π~
L
4. Wave packet probability current density.
1
2~2 π 2
?
mL2
4~2 π 2
mL2
After this mea-
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.
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