Question

Please show all working and clearly label a) and b)

2 A Lloyd's mirror experiment is carried out using a coherent beam of thermal neutrons. In this experiment the neutrons are able to travel over two paths to a detector D, as shown in Figure 1 One path proceeds directly from the neutron source S, through a phase shifter PS to the detector D, whilst the second path requires the neutrons to reflect off a perfect silicon mirror M, before arriving at the detector D Phase Shifier PS Path 1 Thermal Neutron Source Detector Path 2 2e Silicon Mirror M Figure 1: Lloyd's mirror experiment with thermal neutrons The wavefunction incident on the phase shifter PS is represented by Ψ1-AA2, where the amplitude A is considered to be a complex number. The wavefunction incident on the silicon mirror M has the same amplitude and is represented by Ψ,-Ala After passing through the phase shifter PS, the wavefunction y, experiences a phase shift δ, so that the corresponding wavefunction at the detector D is given by Ψ1e". Here δ is a real number Similarly, after reflecting off the silicon mirror M, the wavefunction w2 acquires a phase shift x, whence the corresponding wavefunction at the detector D is v^e*. Write an expression for the probability density when vi and v2 are superposed. Show that the neutron intensity recorded at the detector D is: (a) (b) Hints: You may assume an ideal detector for which the neutron intensity is equal to the probability density The jollowing identities may be required: 2i

Answer 1

Phi = psi_1 e^2 omega t phi' = psi_2 e^(omega t - (f + pi)) phi' = - psi_2 e^i (omega t - f) f rightarrow phase difference b/w 2 beam psi = A/squareroot 2 - A/squareroot 2 e^if psi^* = A/squareroot 2 - A/squareroot 2 e^i f so, neutron, intensity reached at the vector D is I = psi^* psi = A^*/squareroot 2 middot A/squareroot 2 [1 - e^i f][1 - e^- i f] = mod A^2/2 [1 - e^- i f - e^i f + 1] = mod A^2/2 [2 - (e^i f + e^- i f)] = mod A^2/2 [2 - 2 cos] implies mod A^2 [1 - cos f] = mod A^2 [1 - (1 - 2 sin^2 (f/2))] = 2 mod A sin^2 (f/2)