Download Quantum Theory 1 - Home Exercise 9

Quantum Theory 1 - Home Exercise 9
1. Consider a particle moving in a central force. We’ve seen that the differential form of the
angular momentum operators is given by
∂
∂
L̂x = −i~ − sin ϕ
,
− cos ϕ cot θ
∂θ
∂ϕ
∂
∂
L̂y = −i~ cos ϕ
,
− sin ϕ cot θ
∂θ
∂ϕ
∂
L̂z = −i~
.
∂ϕ
(a) Calculate the differential form of L̂+ and L̂− .
(b) Use a direct calculation(integrals over wavefunctions etc.) to calculate the matrix representations of the following operators given that l = 2.
i. L̂x
ii. L̂y
iii. L̂z
iv. L̂+
v. L̂−
vi. L̂2
(c) Repeat the calculation using raising and lowering operators.
2. Consider a particle moving in a potential with spherical symmetry. At time t = 0 the particle
is in a state
ψ(x, y, z) = C (xy + yz + zx) e−α(x
2 +y 2 +z 2
)
(a) Calculate the probability to measure L̂2 = 0.
(b) Calculate the probability to measure L̂2 = 6~2 .
(c) A measurement of L̂2 yields 6~2 . Afterwards, we measure L̂z . What are the possible
values in this measurement? What is the probabilities of measuring any of these values?
(d) What is the probability to measure L̂x = 0?
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3. Consider a symmetrical top with moments of inertia Ix = Iy and Iz . The Hamiltonian of the
top is given by
H=
1 2
1 2
L̂x + L̂2y +
L̂
2Ix
2Iz z
(a) Find the eigenvalues and eigenstates of the Hamiltonian.
(b) Find the expectation value of L̂x + L̂y + L̂z for a state |l, mi.
(c) Given that at time t = 0 the top is at a state |3, 0i, What is the probability that at time
t = 4πIx /~ we will measure L̂x = ~?
4. (a) Can the expression
ω0
~ L̂x L̂z
be a Hamiltonian?
Consider a particle moving in a central force potential. The Hamiltonian is
given by
Ĥ =
ω0 L̂x L̂z + L̂z L̂x
~
(b) Is it possible to measure both the energy and the angular momentum L̂2 of the particle?
(c) Given that the particle has angular momentum l = 1. What are the eigenstates and
eigenvalues of Ĥ?
The particle is prepared in the state
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|ψi (t = 0) = √ (|1, 1i − |1, −1i)
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(d) Find |ψi (t)
(e) At time t > 0 we measure L̂z . Find the possible outcomes and the probabilities of
measuring them.
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