The Quantum Mechanics Solver 15 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 | Solutions 139 The vector a is normalized oo a a e a 2 n 0 a nan n 1 . The expectation value of the number of photons in that state is In o 2V a o d d a d o 2 a 2 . b The time evolution of t is given by t V a e-iw 1 2 i n 2 n 0 - -1 a I 2 2 n n 0 n e-iwi 2 oe-iwi . c The expectation values of the electric and magnetic fields are I hw 0V Ux E ry t 2a cos wt sin kz B r t 0 4 1 hwk0 -2a sin wt cos kz uy . V y d These fields are of the same type as the classical fields considered at the beginning of the problem with s I hw e t - 70v cos ut b t -2a -y sin wt . Given the relation e0 0c2 1 we verify that e t c2kb t and b -ke t . Therefore the expectation values of the field operators satisfy Maxwell s equations. e The energy of the classical field can be calculated using the result of question . Since cos2 ut sin2 ut 1 we find U t hua2. This classical energy is therefore time-independent. The expectation value of HC is HC hu N 1 2 hu a2 1 2 . It is also time independent Ehrenfest s theorem . f For a much larger than 1 the ratio U t HC is close to 1. More generally the expectation value of a physical quantity as calculated for a quantum field in the state o will be close to the value calculated for a classical field such that Ecl r t E r t and Bcl r t B r t. 140 14 Direct Observation of Field Quantization a l 2 M - E 2ha 1 4 M - E ha o x b 1 0 0 o - E 0 Fig. . a Positions of the five first energy levels of Ho. b Positions of the five r-j i i r-rr rr . T î r first energy levels of H Ho W Section The Coupling of the Field with an Atom . One checks that H0 f n - 2A n I h f n H0 e n 2A in j h e ri . . For a cavity which resonates at the atom s frequency . if a the couple of states f n 1 e ri are degenerate. The first five levels of Ho are shown in Fig. . Only the ground state f 0 of the atom field system is non-degenerate. . a The action of W on the basis vectors of Ho is given by W f ri y ny e n 1 if n 1 0 if n 0 W e ri Vn 1 7 f n 1
