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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? 1 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 1 |ψi (t = 0) = √ (|1, 1i − |1, −1i) 2 (d) Find |ψi (t) (e) At time t > 0 we measure L̂z . Find the possible outcomes and the probabilities of measuring them. 2