Basic Theoretical Physics: A Concise Overview P27. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 268 33 Time-dependent Perturbations For f i the Schrodinger equation yields the following result cf t -1 V - 1 h Ji Wfi w Here only the linear terms in V have been considered and the non-resonant terms k V e lwi . with w w have also been neglected vf0 stands for 0 V L 0 Uf V ui By squaring the above result one obtains 2 wfi w t f Ol2 iJf-T W fi W 2 I- J This corresponds to a periodic increase followed by a decrease with the Poincare repetition time At 2n wfi w which is extremely long near a resonance of the denominator. Thus with a source of radiation consisting of n uncorrelated radiators of almost the same frequency wa w . n V e- iwt 2 e a a 1 with random phases r a one obtains the n-fold result of if the frequencies are identical . In contrast if the radiation were coherent . laser radiation one would obtain the n2-fold result. However in that case it makes no sense to interrupt the time-dependent perturbation series as we did after the lowest order. In fact at this point the transition from coherent and reversible quantum mechanics to incoherent and irreversible behavior occurs as in statistical physics Part IV . Thus if one has a continuum of sources of incoherent radiation . with 2 I dwa @7 wa then one obtains as transition rate Wi f transition probability i f divided by the time t Wi f lim Cf t 2n v 2 wfi . 1 t- t h2 fi I fi Selection Rules 269 In the above proof we have used the identity sin2 f z x lim ---------- I 2nd wJi w . t f-tf tJ The matrix elements have been incoherently averaged as expressed by the bar in . Equation describes transitions from a discrete energetically lower level i to an energetically higher level f by induced absorption of continuous radiation of frequency w with an incoherent density qy w . See also Fig. below. Conservation of energy w wj wi is explicitly given by the -function in the above formal correspondence. By permutation of f and i and the simultaneous replacement w w one
