tailieunhanh - The Quantum Mechanics Solver 6

The Quantum Mechanics Solver 6 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | 42 3 Neutron Interferometry Comparing the values bn of question with this experimental result and recalling the result of a measurement of gx for these values explain why this proves that the state vector of a spin-1 2 particle changes sign under a rotation by an odd multiple of 2n. Solutions Section Neutron Interferences . The beams ABDC2 and ACDC2 interfere. Omitting the propagation factors one has at C2 an amplitude A2 a2p p3 p a2 p2 . Similarly for ABDC3 and ACDC3 A3 2ap2 . The intensities at the two counters are I2 R - 4R2T I3 4R2T . . When there is a phase shift 5 in C the above expressions get modified as follows A2 a2 pe16 p3 p a2e16 p2 A3 ap2 1 e16 . The intensities become I2 R - 2R2T 1 cos 5 I3 2R2T 1 cos 5 . The fact that I2 I3 does not depend on the phase shift 5 is a consequence of the conservation of the total number of particles arriving at D. Section The Gravitational Effect . This results from elementary trigonometry. a Since there is no recoil energy of the silicon atoms to be taken care of the neutron total energy kinetic potential is a constant of the motion in all the process. This energy is given by EaC p2 2M and Ebd p Ap 2 2M MgH sin p hence Ap M2gH sin p p . b The velocity - 2gH is of the order of m s and the neutron velocity is v h MX 2700 m s. The change in velocity Av is therefore very small Av gH v 2 x 10 -4 m s for p n 2. Solutions 43 . a The gravitational potential varies in exactly the same way along AB and CD. The neutron state in both cases is a plane wave with momentum p h X just before A or C. The same Schrodinger equation is used to determine the wave function at the end of the segments. This implies that the phases accumulated along the two segments AB and CD are equal. b When comparing the segments AC and BD the previous reasoning does not apply since the initial state of the neutron is not the same for the two segments. The initial state is exp ipz h for AC and exp i p Ap z h for BD.