Time Dependence in Quantum Mechanics John S. Briggs ∗ and Jan M. Rost † Institute for Advanced Study, Wallotstr. 19, D-14193 Berlin, Germany (February 1999) It is shown that the time-dependent equations (Schr¨odinger and Dirac) for a quantum system can be always derived from the time-independent equation for the larger object of the system interacting with its environment, in the limit that the dynamical variables of the environment can be treated semiclassically. The time which describes the quantum evolution is then provided parametrically by the classical evolution of the environment variables. The method used is a generalization of that known for a long time in the field of ion-atom collisions, where it appears as a transition from the full quantum mechanical perturbed stationary states to the impact parameter method in which the projectile ion beam is treated classically. PACS numbers: 3.65 Bz, 34.10, 3.65 Sq I. INTRODUCTION where {x} are the system variables. Notwithstanding the intuitive appeal of the TDSE there are several problems Originally Schr¨odinger [1] proposed his wave equation connected with its use, for example; in time independent form as an eigenvalue equation for (A) Although many authors have contrived to define the time-independent Hamilton operator, i.e. one [3], there is no simple obvious definition of a (E − H)Ψ=0, (1) Hilbert space operator corresponding to time. The ”usual replacement” the time-independent Schr¨odinger equation (TISE). Sub- sequently, this was generalized to consider quantum sys- ∂ E → i¯h (4) tems as being described by state vectors in Hilbert ∂t space and observables by Hermitian operators. Pairs of canonically conjugate operators satisfy Heisenberg is unusual in that E is the eigenvalue of the op- commutation rules and thereby fulfill corresponding erator H, but i¯h∂/∂t is not the eigenvalue of an Heisenberg uncertainty relations. In a later paper [2] operator. Schr¨odinger introduced a time-dependent Schr¨odinger equation (TDSE) and this version has found much wider (B) Following from (A), there is no energy-time un- application than the TISE even for time-independent certainty relation, although time, as measured by Hamiltonians. On the one hand this is due to the tech- clocks, is arguably an observable and countless nical reason that it is easier to solve the initial value books refer to the “energy-time” uncertainty rela- problem of the TDSE than the boundary value problem tion. of the TISE. On the other hand there is the deeper rea- (C) The TDSE admits time-dependent Hamiltonians. arXiv:quant-ph/9902035v1 9 Feb 1999 son that our experience based on the classical world still Time dependent potentials are often introduced in conditions us to think in terms of physical processes pro- an ad hoc way and lead to non-Hermitian Hamilto- ceeding from some initial state and developing in time to nians. This implies a loss of norm of the quantum some final state. This is despite the fact that quantum wavefunction. i.e. to an ad hoc loss of particles mechanics teaches that the state Ψ of a closed system from the quantum system. can be written as a time-independent superposition (D) The TDSE is simply postulated, with the tacit as- Ψ= cnΦn (2) sumption that the parameter called time is to be Xn identified with the classical time. There is no proof of states of the same total energy but different internal of this. For example, in the case that HS is time- configurations where the probability of measurement of independent, corresponding to an isolated conser- 2 vative system, the simple phase transformation each possible state is given by |cn| . Almost all books on quantum mechanics simply postu- ˜ late a TDSE for a quantum system S with Hamiltonian ψS (x, t) = exp(i/¯hES t)ψS (x) (5) HS in (3) leads to the TISE ∂ i¯h − HS ψ˜S (x, t)=0, (3) ∂t (ES − HS )ψS (x)=0, (6) 1 where ES is well-defined as an eigenvalue of HS . ter field quantum mechanically, in a procedure that is Clearly, in this procedure (or its inverse), the similar to that used in the case of atomic collisions [7]. ”time” t merely appears as a mathematical param- Starting with the TISE of (1) for E ⊗ S we will de- eter. rive the TDSE (3) for S. Thereby we will show that the parametric time derivative arises from the expectation Here we will show that these difficulties can all be over- values of the environment operators, thus resolving prob- come in that the TDSE for a quantum system S can be lem (A) and eliminating the need for the replacement derived from the TISE for the larger object comprising (4). Similarly, we can show that the energy-time ”uncer- the system S and its quantum environment E. In the tainty relation” for S arises from true (operator based) limit that the environment is so large (precisely what uncertainty relation for E, explaining problem (B). Fur- this means depends upon the object under discussion) thermore, it will be shown that the time-dependence of that E is practically unchanged in its interaction with HS (t) arises in a well defined way from the interaction of S, it can be treated semiclassically. In this limit the S with E (problem (C)) and that the time which arises variables of the environment undergo a time develop- is precisely the time describing the classical motion of E ment described by classical equations involving a classical i.e. the classical environment provides the clock for the time parameter as measured by a clock. In the same ap- quantum system (problem (D)). The further approxima- proximation the system S develops in time governed by tion that the interaction of S with E can be neglected the TDSE with an effective time-dependent Hamiltonian gives a time-independent Hamiltonian HS appropriate whose time dependence arises from the interaction with to a non-interacting (closed) quantum system and the E via the implicit time-dependence of the classical en- TDSE (3) reduces to the new TISE (6) for S alone. vironment variables. In this way time always arises in It is clear that the procedure can now be continued in quantum mechanics as an externally defined classical pa- that S, described by the TISE (6), can be considered as rameter and time-dependent Hamiltonians from the in- composed of S′ and E′ to define time and a TDSE for S′. teraction with a classical environment. The most im- The extent to which the subdivision is valid depends on portant example of the interaction of a quantum sys- the accuracy with which the dynamics of E′ may be ap- tem S with an environment is the act of observation proximated (semi-)classically. More pragmatically, one or measurement, when time is defined by the classical might say that the position of the interface between the (macroscopic) measuring device. Furthermore, since all quantum and the classical worlds depends upon the de- measurements ultimately involve the detection of charged gree of precision set (or achieved) by the measurement. particles, photons, or heat (phonons) these are the types In section II the interaction of a quantum system with of environment we shall consider. Their classical mo- a material environment is considered. The TDSE is de- tion is described by Newton or Maxwell equations and rived by generalizing the procedure due to Briggs and hence the time parameter introduced into the TDSE for Macek [4] who considered the particular case of an atom the quantum system is identical with that entering the interacting with a particle beam. Then it is shown classical equations. how the energy-time uncertainty relation arises. In sec- In deriving a TDSE from the TISE of a composite sys- tion III the general procedure is illustrated by the three tem, we will follow closely the development of Briggs and generic examples of a system S interacting with a parti- Macek [4], who considered the particular case of a beam cle beam or with a set of quantum oscillators (photons or of ions (environment) interacting with a target atom phonons) as environment E. Finally, the same procedure (quantum system). In this particular case they showed can be applied to transform the time-independent Dirac how the initially time-independent equation for the cou- equation (TIDE) into the time-dependent Dirac equation pled ion-atom reduces to a time-dependent equation for (TDDE). Here it is interesting to observe that the time the atom alone, in the limit that the ion is considered component of the spacetime of the quantum system actu- to move along a classical trajectory. In fact the method ally arises from the implicit time variation of the classical is much older in origin in atomic collision physics and environment variables, i.e. again it is this variation that can be traced through the ”perturbed stationary states” provides the clock for the quantum system. (PSS) method of Mott and Massey [5] to the original 1931 paper of Mott [6], where he showed the essential equiv- alence of the time-independent PSS and time-dependent II. THE EMERGENCE OF TIME impact parameter approaches in the limit of high beam velocities, with the time defined by the variables of the We begin by decomposing the total Hamiltonian H for particle beam. The PSS method is a generalization of the the large object in (1) into adiabatic Born-Oppenheimer method as applied to sta- tionary states of molecules. Interestingly, this method H = HE + HES + HS (7) of molecular physics has become very popular recently in defining time in quantum gravity.