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Basic Theoretical Physics: A Concise Overview P25

Basic Theoretical Physics: A Concise Overview P25. 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. | 246 28 Abstract Quantum Mechanics Algebraic Methods Up till now we have used Schrodinger s picture . The transitions to the other representations ii the Heisenberg or iii the Dirac interaction representations are described in the following. The essential point is that one uses the unitary transformations corresponding to the so-called timedisplacement operators U t t0 . In addition we shall use indices S H and I which stand for Schrodinger Heisenberg and interaction respectively. a In the Schrodinger picture the states are time-dependent but in general not the operators e.g. position operator momentum operator . . We thus have . . . . . S t JJ t t0 s to . Here the time-displacement operator U t t0 is uniquely defined according to the equation -1t HsU Mo i dt i.e. the Schrodinger equation and converges 1 for t t0. Here t0 is fixed but arbitrary. b In the Heisenberg picture the state vectors are constant in time W H S t0 whereas now the operators depend explicitly on time even if they do not in the Schrodinger picture i.e. we have in any case Ah t Jj t t0 As t U t t0 . 28.12 If this formalism is applied to a matrix representation AH j k t H j Ah t H k one obtains quasi by the way Heisenberg s matrix mechanics. c In Dirac s interaction picture the Hamilton operator H Hs is decomposed into an unperturbed part and a perturbation Hs Ho Vs where H0 does not explicitly depend on time whereas Vs can depend on t. One then introduces as unperturbed time-displacement operator the unitary operator Ho t to h T T t i i U0 t t0 e 28.3 Unitary Equivalence Change of Representation 247 and transforms all operators only with Uo i.e. with the definition Ai t U t t0 As t Uo t to . 28.13 Ai t is thus a more or less trivial modification of As t although the time dependence is generally different. In contrast the time-displacement of the state vectors in the interaction picture is more complicated although it is already determined by 28.13 plus the postulate that the physical quantities .