Answer Happy

lub) – (in+) 12–)2 – 12–)_]v+)2), In 5. Angular momentum Take as given the formulae at the foot of the question. (a) For a spin- particle in a p-state, with wavefunctions Ym(t) = f(r)Y1,00(0,6). where m = -1,0,1, find the simultaneous eigenstates of ſ2 and Ìz, where I is the total angular momentum operator of the particle. [8 marks) (b) Suppose that two spin-neutrons are produced in a singlet state at a source: = 1 +. In where In+), denotes a single particle state where particle i is spin-up along a particular axis n. Suppose that the neutrons fly away in opposite directions and that their individual spins are analysed by Stern-Gerlach devices located at A and B. Suppose further that the device at A is set along the z-direction and the device at B set along the r-direction. If measurement at A yields spin up in the z-direction, show that the probabilities of measuring the particle at B to be spin up or spin down along the r-direction are both 1/2. [4 marks) (c) Let în be an arbitrary unit vector and let S = ĈS,+ģS,+2S, be the spin operator in vector form. Compute the eigenvectors and eigenvalues of the observable N-S. Hint: Assume without loss of generality that ñ lies in the rz plane, such that Ñ = (sin 0, 0, cos). [8 marks] The non-dimensional lowering operator acting on a generic angular-momentum eigenstate: Î-16, m) - e(l +1) – m(m – 1)|e, m - 1) • The Pauli matrices: 0 1 1 0 1 0 0 -1 • Eigenstates for spin-1/2 particles (spin measured along the z-axis): 1 1 1 2+) 1 Eigenstates for spin-1/2 particles (spin measured along the z-axis): () -- () 1 - -0 -- (0) 12+) 12-> 0 1 000