Question

Question 5 [2 + 2 + 2 + 3 + 3 = 12 marks ] (a) Briefly explain why we cannot find simultaneous eigenfunctions of Lg, Ly and Lz. An electron in a hydrogen atom is in the n = 2 state. Ignoring spin, write down the list of possible quantum numbers {n, l, m}. (b) For two qubits briefly explain, giving examples, the difference between a product state and an entangled state (c) Consider a system of identical bosons and a system of identical fermions. Briefly explain the difference in the ground state energy configurations. (d) Consider two distinguishable particles in 1D which exist in single-particle (normalised) wave functions Va(xı) and (x2). Write down the combined two-particle wave function and show that the expectation value of the squared separation is given by: ((x1 -x2)2) = (x2)a + (x2)b - 2(x)a(x)b, where (x")a,b denotes an expectation value of r" with respect to Va,b (e) The quantum logic gates X, Z, H and CNOT act on qubit states as follows: 1) -» |0) j0)-1), Х: Z 0) -> |0), |1) -» -|1) )(1/V2)(I0) - |1)) j0) (1/2)(0 00) |00), |01) -» |01), |10) - |11), 11)> |10) 1)), Н: CNOT where in the CNOT gate above the control qubit state corresponds to the first entry in the ket, and the target qubit state corresponds to the second entry. At t = 0 a quantum computer comprising two qubits starts out in the state where both qubits are initialised in the |0) state, i.e. |V(0)) = |00). Draw a circuit, carefully labelling the qubits and gates, which describes the following sequence of gates: H on qubit 1, X on qubit 1, Z on qubit 2, CNOT (control = qubit 1, target = qubit 2), Z on qubit 1. Find the final state of the two qubits.

Answer 1

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