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The Quantum Mechanics Solver 17

The Quantum Mechanics Solver 17 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 | 16.5 Solutions 161 c If wq w wi we have to a good approximation the differential system i j e- 3- i j- - 3 whose solution is indeed ß t ß to cos 1 t2 to - i e ß t0 sin 1 t2 to . d Defining i ti - to 2 x 1 - to 2 ô w t1 to 2 we obtain a t1 e 1Xß t1 e 1X a t0 cosß ia t0 e lj sinß a - t1 elxß t1 e 1X a - t0 cos ß ia t0 e 1wi sin ß and therefore Ze 1X cos ß y i e16 sin ß i e 16 sin ß e1X cos ß Section 16.2 Ramsey Fringes 16.2.1. We assume fr n 4 the initial conditions are a t0 0 a - to 1. At time t1 the state is ß t1 -1 ie 16 n eX n In other words a . t1 ie 15 V2 a - t1 e1x V2 and P 1 2. 16.2.2. We set T D v. The neutron spin precesses freely between the two cavities during time T and we obtain a t0 A a - t0 .ie 16 e 1al T 2 e1Xe 1w T 2 16.5 16.2.3. By definition t 0 t1 T and t 1 2t1 t0 T therefore the transition matrix in the second cavity is e 1X cos ß ie16 sin ß ie 16 sin ß e1X cos ß1 with wi ti t0 2 x X w 1 o 2. Only the parameter ô is changed into 5 w t1 10 2 w 3t1 2T t0 2 . 162 16 The Quantum Eraser 16.2.4. The probability amplitude for detecting the neutron in state after the second cavity is obtained by i applying the matrix U to the vector 16.5 ii calculating the scalar product of the result with n . We obtain a t1 1 i ie-i x u T 2 ie -i - x- T 2 2 V Since 3 X wti 3 _ x 2 3ti 2T to _ ti to w ti T we have a t e-iw il T 2 e-i u -u T 2 ei w -w T 2 16.6 Therefore the probability that the neutron spin has flipped in the two-cavity system is P a ti 2 cos2 2 0 T With the approximation of Sect. 1.2c the probability for a spin flip in a single cavity is independent of w and is equal to 1 2. In contrast the present result for two cavities exhibits a strong modulation of the spin flip probability between 1 e.g. for w w0 and 0 e.g. for w w0 T n . This modulation results from an interference process of the two quantum paths corresponding respectively to a spin flip in the first cavity and no flip in the second one no flip in the first cavity and a spin flip in the second .