Question

[ 2 + 2 + 2 + 3 + 3 = 12 marks ] Question 5 (a) Briefly explain why we cannot find simultaneous eigenfunctions of Lt, L, 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(x1) and (x2). Write down the combined two-particle wave function and show that the expectation value of the squared separation is given by: ((r1 -x2)2) = (x2)a + (x2)b - 2(x)a(x)b, where (x")a,b denotes an expectation value of x" with respect to Va,b (e) The quantum logic gates X, Z, H and CNOT act on qubit states as follows: 0) - |1), |1) -> |0) X (ol(ol 1) 1) 0)(1//2)(l0) + |1)), |1) -> (1//2)(0) - |1)) Z H CNOT |00) 00), 01)01), |10) -+ |11), |11)-+ |10), 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. |T(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.