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

Lx = −i~(− sinϕ

∂

∂θ− cosϕ cot θ

∂

∂ϕ

),

Ly = −i~(

cosϕ∂

∂θ− sinϕ cot θ

∂

∂ϕ

),

Lz = −i~(∂

∂ϕ

).

(a) Calculate the differential form of L+ and L−.

(b) Use a direct calculation(integrals over wavefunctions etc.) to calculate the matrix repre-

sentations of the following operators given that l = 2.

i. Lx

ii. Ly

iii. Lz

iv. L+

v. L−

vi. L2

(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(x2+y2+z2)

(a) Calculate the probability to measure L2 = 0.

(b) Calculate the probability to measure L2 = 6~2.

(c) A measurement of L2 yields 6~2. Afterwards, we measure Lz. 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 Lx = 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

2Ix

(L2x + L2

y

)+

1

2IzL2z

(a) Find the eigenvalues and eigenstates of the Hamiltonian.

(b) Find the expectation value of Lx + Ly + Lz for a state |l,m〉.

(c) Given that at time t = 0 the top is at a state |3, 0〉, What is the probability that at time

t = 4πIx/~ we will measure Lx = ~?

4. (a) Can the expression ω0~ LxLz be a Hamiltonian?

Consider a particle moving in a central force potential. The Hamiltonian is

given by

H =ω0

~

(LxLz + LzLx

)(b) Is it possible to measure both the energy and the angular momentum L2 of the particle?

(c) Given that the particle has angular momentum l = 1. What are the eigenstates and

eigenvalues of H?

The particle is prepared in the state

|ψ〉 (t = 0) =1√2

(|1, 1〉 − |1,−1〉)

(d) Find |ψ〉 (t)

(e) At time t > 0 we measure Lz. Find the possible outcomes and the probabilities of

measuring them.

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