Question

[ 2 + 3 + 3 + 5 = 13 marks ] Question 4 (a) For the case of two qubits, briefly explain the main difference between product states and entangled states. Provide one example of each. (b) Show that the identity between spin angular momentum operators S+S- = S2- S2 ± hS, holds. Data: S S tiSy, [Se, Sy] = ihS2, [Sy, S2] = ihS, [S2, S = ihSy. (c) An s = 1/2 particle at t =0 is prepared in the following state x with respect to the eigen-spinors of S2: x= A 2 (E)r= where A is a normalisation constant. Imagine many copies of the state x are prepared and measured. Describe the possible outcomes of measurements of S2 and their probabilities (d) Show that for a normalised trial state, Vtrial), the ground state energy, Eg.s., of a Hamiltonian, H, is bounded as (trial|H\btrial). Hence, prove that the first order perturbation theory estimate of the Eg.S. ground state energy never under-estimates the true ground state energy.

Answer 1

2isy Let's eolnlate One me tem STS - isxy ysx+ S 2 2 Sivilay isSy-isy 2

5x Sy 2 2 NG We kngs, Sxy S+ ts 4 X A JA1 A

치고 2 m t p agindlal a we meaile S (2) eauld ut72 wth We with pto babality Ja1 and babildy So (

avefunstin ind7 am be e osa line d anlkination n the But the Loumd Atk endy is, by defimilion EsEn pmallest eizenalue Guffiths, uantum-mechanies, Pas: 32 3 Volviatisnnd Prumeiple Souce