Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 17
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For time-independent Hamiltonians with Û (tb , ta ) given by(1.232), the proof of (1.255) is trivial. In the general case (1.253), it follows fromthe simple manipulation valid for tb > ta :iT̂ exp −h̄"Ztbt0iĤ(t) dt T̂ exp −h̄Zt0ta!Ĥ(t) dt!#i Z tbi Z t0= T̂ exp −Ĥ(t) dt exp −Ĥ(t) dth̄ t0h̄ tai= T̂ exp −h̄Ztbta(1.256)Ĥ(t) dt .b) UnitarityThe expression (1.253) for the time evolution operator Û(tb , ta ) was derived only forthe causal (or retarded ) time arguments, i.e., for tb later than ta . We may, however,define Û (tb , ta ) also for the anticausal (or advanced ) case where tb lies before ta . Tobe consistent with the above composition law (1.255), we must have−1Û (tb , ta ) ≡ Û(ta , tb ) .(1.257)Indeed, when considering two states at successive times|Ψ(ta )i = Û (ta , tb )|Ψ(tb )i,(1.258)H.
Kleinert, PATH INTEGRALS391.7 Properties of Time Evolution Operatorthe order of succession is inverted by multiplying both sides by Û −1 (ta , tb ):|Ψ(tb )i = Û (ta , tb )−1 |Ψ(ta )i,tb < ta .(1.259)The operator on the right-hand side is defined to be the time evolution operatorÛ (tb , ta ) from the later time ta to the earlier time tb .If the Hamiltonian is independent of time, with the time evolution operator beingÛ (ta , tb ) = e−i(ta −tb )Ĥ/h̄ ,ta > tb ,(1.260)tb < ta .(1.261)the unitarity of the operator Û (tb , ta ) is obvious:−1Û † (tb , ta ) = Û(tb , ta ) ,Let us verify this property for a general time-dependent Hamiltonian. There, adirect solution of the Schrödinger equation (1.163) for the state vector shows thatthe operator Û (tb , ta ) for tb < ta has a representation just like (1.253), except for areversed time order of its arguments.
One writes this in the form [compare (1.253)]Û (tb , ta ) = Tˆ expi Z tbĤ(t) dt ,h̄ ta(1.262)where Tˆ denotes the time-antiordering operator, with an obvious definition analogous to (1.242), (1.243). This operator satisfies the relationhT̂ Ô1 (t1 )Ô2 (t2 )i†= Tˆ Ô2† (t2 )Ô1† (t1 ) ,(1.263)with an obvious generalization to the product of n operators. We can thereforeconclude right away thatÛ † (tb , ta ) = Û (ta , tb ),tb > ta .(1.264)With Û (ta , tb ) ≡ Û (tb , ta )−1 , this proves the unitarity relation (1.261), in general.c) Schrödinger equation for Û (tb , ta )Since the operator Û (tb , ta ) rules the relation between arbitrary wave functions atdifferent times,|Ψ(tb )i = Û(tb , ta )|Ψ(ta )i,(1.265)the Schrödinger equation (1.228) implies that the operator Û (tb , ta ) satisfies thecorresponding equationsih̄∂t Û(t, ta ) = Ĥ Û (t, ta ),ih̄∂t Û (t, ta )−1−1= −Û (t, ta ) Ĥ,(1.266)(1.267)with the initial conditionÛ (ta , ta ) = 1.(1.268)401.81 FundamentalsHeisenberg Picture of Quantum MechanicsThe unitary time evolution operator Û (t, ta ) may be used to give a different formulation of quantum mechanics bearing the closest resemblance to classical mechanics.This formulation, called the Heisenberg picture of quantum mechanics, is in a wayscloser related to to classical mechanics than the Schrödinger formulation.
Manyclassical equations remain valid by simply replacing the canonical variables pi (t)and qi (t) in phase space by Heisenberg operators, to be denoted by pHi (t), qHi (t).Originally, Heisenberg postulated that they are matrices, but later it became clearthat these matrices had to be functional matrix elements of operators, whose indicescan be partly continuous. The classical equations hold for the Heisenberg operatorsand as long as the canonical commutation rules (1.93) are respected at any giventime.
In addition, qi (t) must be Cartesian coordinates. In this case we shall always use the letter x for the position variable, as in Section 1.4, rather than q, thecorresponding Heisenberg operators being xHi (t). Suppressing the subscripts i, thecanonical equal-time commutation rules are[pH (t), xH (t)] = −ih̄,[pH (t), pH (t)] = 0,(1.269)[xH (t), xH (t)] = 0.According to Heisenberg, classical equations involving Poisson brackets remainvalid if the Poisson brackets are replaced by i/h̄ times the matrix commutators atequal times.
The canonical commutation relations (1.269) are a special case of thisrule, recalling the fundamental Poisson brackets (1.25). The Hamilton equations ofmotion (1.24) turn into the Heisenberg equationsi[H , p (t)] ,h̄ H Hi[H , x (t)] ,ẋH (t) =h̄ H H(1.270)HH ≡ H(pH (t), xH (t), t)(1.271)ṗH (t) =whereis the Hamiltonian in the Heisenberg picture. Similarly, the equation of motion forarbitrary observable function O(pi(t), xi (t), t) derived in (1.20) goes over into thematrix commutator equation for the Heisenberg observableOH (t) ≡ O(pH (t), xH (t), t),(1.272)namely,di∂OH = [HH , OH ] + OH .dth̄∂tThese rules are referred to as Heisenberg’s correspondence principle.(1.273)H.
Kleinert, PATH INTEGRALS411.8 Heisenberg Picture of Quantum MechanicsThe relation between Schrödinger’s and Heisenberg’s picture is supplied by thetime evolution operator. Let Ô be an arbitrary observable in the Schrödinger descriptionÔ(t) ≡ O(p̂, x̂, t).(1.274)If the states |Ψa (t)i are an arbitrary complete set of solutions of the Schrödingerequation, where a runs through discrete and continuous indices, the operator Ô(t)can be specified in terms of its functional matrix elementsOab (t) ≡ hΨa (t)|Ô(t)|Ψb (t)i.(1.275)We can now use the unitary operator Û (t, 0) to go to a new time-independent basis|ΨH a i, defined by|Ψa (t)i ≡ Û (t, 0)|ΨH a i.(1.276)Simultaneously, we transform the Schrödinger operators of the canonical coordinatesp̂ and x̂ into the time-dependent canonical Heisenberg operators p̂H (t) and x̂H (t) viap̂H (t) ≡ Û (t, 0)−1 p̂ Û(t, 0),x̂H (t) ≡ Û (t, 0)−1 x̂ Û (t, 0).(1.277)(1.278)At the time t = 0, the Heisenberg operators p̂H (t) and x̂H (t) coincide with the timeindependent Schrödinger operators p̂ and x̂, respectively.
An arbitrary observableÔ(t) is transformed into the associated Heisenberg operator asÔH (t) ≡ Û (t, ta )−1 O(p̂, x̂, t)Û (t, ta )≡ O (p̂H (t), x̂H (t), t) .(1.279)The Heisenberg matrices OH (t)ab are then obtained from the Heisenberg operatorsÔH (t) by sandwiching ÔH (t) between the time-independent basis vectors |ΨH a i:OH (t)ab ≡ hΨH a |ÔH (t)|ΨH b i.(1.280)Note that the time dependence of these matrix elements is now completely due tothe time dependence of the operators,ddOH (t)ab ≡ hΨH a | ÔH (t)|ΨH b i.(1.281)dtdtThis is in contrast to the Schrödinger representation (1.275), where the right-handside would have contained two more terms from the time dependence of the wavefunctions. Due to the absence of such terms in (1.281) it is possible to study theequation of motion of the Heisenberg matrices independently of the basis by considering directly the Heisenberg operators.
It is straightforward to verify that they doindeed satisfy the rules of Heisenberg’s correspondence principle. Consider the timederivative of an arbitrary observable ÔH (t),dÔ (t) =dt H!d −1Û (t, ta ) Ô(t)Û(t, ta )dt!!∂d−1−1+ Û (t, ta )Ô(t) Û (t, ta ) + Û (t, ta )Ô(t)Û (t, ta ) ,∂tdt421 Fundamentalswhich can be rearranged as"!#d −1Û (t, ta ) Û(t, ta ) Û −1 (t, ta )Ô(t)Û (t, ta )(1.282)dt!ihd∂−1−1−1+ Û (t, ta )Ô(t)Û(t, ta ) Û (t, ta ) Û (t, ta ) + Û (t, ta )Ô(t) Û(t, ta ).dt∂tUsing (1.266), we obtain!i∂di h −1Û Ĥ Û , ÔH + Û −1ÔH (t) =Ô(t) Û .dth̄∂t(1.283)After inserting (1.279), we find the equation of motion for the Heisenberg operator:iih∂dÔH (t) =ÔĤH , ÔH (t) +dth̄∂t!(t).(1.284)HBy sandwiching this equation between the complete time-independent basis states|Ψa i in the Hilbert space, it holds for the matrices and turns into the Heisenbergequation of motion.
For the phase space variables pH (t), xH (t) themselves, theseequations reduce, of course, to the Hamilton equations of motion (1.270).Thus we have shown that Heisenberg’s matrix quantum mechanics is completelyequivalent to Schrödinger’s quantum mechanics, and that the Heisenberg matricesobey the same Hamilton equations as the classical observables.1.9Interaction Picture and Perturbation ExpansionFor some physical systems, the Hamiltonian operator can be split into two contributionsĤ = Ĥ0 + V̂ ,(1.285)where Ĥ0 is a so-called free Hamiltonian operator for which the Schrödinger equationĤ0 |ψ(t)i = ih̄∂t |ψ(t)i can be solved, and V̂ is an interaction potential which perturbsthese solutions slightly. In this case it is useful to describe the system in Dirac’sinteraction picture.
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