ls2_unit_2

ls2_unit_2 - THE INTERACTION OF RADIATION AND MATTER:...

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THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 12 R. Victor Jones, March 7, 2000 II. R EVIEW OF B ASIC Q UANTUM M ECHANICS : D YNAMIC B EHAVIOR OF Q S YSTEMS : Q UANTUM M ECHANICAL E QUATIONS OF M OTION : To this point our review, we have been concerned with describing the states of a system at one instant of time. The complete dynamical theory must describe, of course, connections between different instants of time. "When one makes an observation on the dynamical system, the state of the system gets changed in an unpredictable way, but in between observations causality applies, in quantum mechanics as in classical mechanics, and the system is governed by equations of motion which make the state at one time determined the state at a later time." 10 Thus, it is only the disturbance caused by the interaction of a system with a measuring device that makes the system’s behavior cease to be strictly causal. S CHRÖDINGER E QUATION OF M OTION : Consider the time evolution of a particular state of an undisturbed system. To deal with such a dynamical system, we need a linear operator of the form 11 ψ t ( ) = T t , t 0 ( ) ψ t 0 ( ) [ II-1 ] 10 From Section 27 of P. A. M. Dirac, The Principles of Quantum Mechanics (Revised fourth edition), Oxford University Press (1967). 11 This is first member of a class of “displacement “ operators that we may treat in a similar fashion. See Section 25 of P. A. M. Dirac, The Principles of Quantum Mechanics (Revised fourth edition), Oxford University Press (1967).
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THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 13 R. Victor Jones, March 7, 2000 Passing to the limit t t 0 , this operator yields a related linear operator for the time derivative of the state vector with respect to t 0 d dt 0 ψ t 0 ( ) = lim t t 0 T t , t 0 ( ) 1 t t 0 Χ Ψ Ω Ϋ ά έ ψ t 0 ( ) = Op t 0 ( ) ψ t 0 ( ) [ II-2a ] If it is postulated that t ( ) = i h ( ) 1 H t ( ) where H t ( ) is the total Hamiltonian (energy) of the system, 12 we obtain the Schrödinger equation of motion in abstract form -- viz. i h d ψ t ( ) = H t ( ) ψ t ( ) [ II-2b ] Equations [ II-1 ] and [ II-2b ] are consistent if i h d T t , t 0 ( ) = H t ( ) T t , t 0 ( ) . [ II-3 ] In the Schrödinger representation , the abstract Schrödinger equation of motion becomes 13 i h t ψ r r , t ( ) = r r H t ( ) r r ψ r r , t ( ) [ II-4 ] H EISENBERG E QUATION OF M OTION : In the Schrödinger picture , as outlined above, we picture the states of the undisturbed motion by associating each state with a moving ket, the state at any time 12 There are two justifications of this postulate: (a) analogy with classical mechanics (see Equation [ II-6 ]) and (b) relativistic invariance vis-a-vis Equation [ I-20 ]. 13 Since ψ t ( ) = d r r r r r r ψ t ( ) = d r r r r ψ r r , t ( ) ∫∫∫
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THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 14 R. Victor Jones, March 7, 2000 corresponding to the ket at that time. In the Heisenberg picture a unitary transformation is applied which brings to rest the kets corresponding to states of undisturbed motion. In this picture, the appropriate equation of motion is one describing the motion or evolution of linear operators (dynamic variables) -- viz. Op t = T 1 Op T
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ls2_unit_2 - THE INTERACTION OF RADIATION AND MATTER:...

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