Time Dependence in Quantum Mechanics

Home , Closed system

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Hamilto- implies non-Hermitian to This lead nians. and way in hoc introduced ad often an are potentials dependent Time countless rela- uncertainty and “energy-time” observable the tion. to an refer by arguably books measured is as time, clocks, although relation, certainty operator. aiesse,tesml hs transformation phase conser- simple the isolated system, an vative to corresponding independent, n 3,teei osml biu ento fa of The definition time. replacement” obvious to ”usual corresponding simple operator no space is Hilbert there [3], one suuuli that erator in unusual is { x } r h ytmvrals owtsadn the Notwithstanding variables. system the are H ia)fraqatmsse can system quantum a for Dirac) beto h ytminteracting system the of object r but , † ψ ˜ S sdi eeaiaino that of generalization a is used ( ,t x, ( ,Germany n, E i S ¯ h∂/∂t iomn a etreated be can vironment exp( = ) − rvddparametrically provided satasto rmthe from transition a as E E H → S steegnau fteop- the of eigenvalue the is ) sntteegnau fan of eigenvalue the not is ψ i i/ ¯ h S ¯ hE ∂t ( ∂ x 0 = ) S t ) ψ S , ( x (5) ) H S stime- is (6) (4) 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 stationary 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. Here time is intro- with the Hamiltonians HE for the environment and HS duced into the time-independent Wheeler-de Witt equa- for the quantum system. For convenience we assume a tion by treating gravitation semi-classically but the mat- coordinate representation for E with a standard form of

2 HE = K + VE where the potential energy is a function of χ [US (R) − HS − HES ] ψ(x, R) coordinates only, VE = VE (R) and the kinetic energy is ¯h ∂ψ(x, R) = C , (14) written in mass scaled coordinates R = (R1, R2,...)asa i i ∂R sum over all degrees of freedom, i =1,...,n: Xi i

2 n where the operator C is given by ¯h ∂2 i K = − 2 . (8) 2M ∂Ri 1 ¯h ∂ 1 ¯h ∂χ Xi Ci = χ + . (15) 2M i ∂Ri M i ∂Ri Almost all relevant environments can be cast into this form as will be illustrated in section III. In (7) HES de- In accordance with the asymmetry condition (10) we have scribes the coupling between S and E. However, as a assumed that consequence of the environment being “large” compared to the quantum system, the coupling is asymmetric in the [HES ,χ]=0. (16) sense that the state χ of the environment E depends neg- Note that (14) is an equation for the wavefunction ψ ligibly on the variables {x} of the system S while the sys- while χ, already fixed in (12) acts like a potential, i.e. an tem state ψ depends on the environment variables {R}. operator. Accordingly, we write the total wavefunction Ψ in (1) as With the form of (13) for χ we can write for the operator C of (15) Ψ(x, R)= χ(R)ψ(x, R). (9) i 1 ¯h ∂ 2 ∂A 1 ∂W Having defined the wavefunction of the system we can Ci = χ + + (17) express what a “large” environment means in terms of 2M i ∂Ri A ∂Ri M ∂Ri an asymmetry condition. It defines and distinguishes, in 1 where (∂W/∂R )M − = P /M = dR /dt is the classical the decomposition of H, environment E and system S i i i momentum vector of the environment E. The most im- through the respective energy expectation values by portant step to turn (14) into a TDSE for the system S and its wavefunction ψ is to keep only the term of lowest hχ|HE |χiR ≡ EE ≫ ES ≡ hχ|US |χiR, (10) order in ¯h in (17) which reduces the operator Ci to where 1 ∂W Pi dRi Ci ≈ χ = χ = χ . (18) US (R)= hψ(x, R)|H − VE |ψ(x, R)ix. (11) M ∂Ri M dt

A more detailed discussion of the requirements for the va- From this approximation for Ci emerges the classical time on the right hand side of (14) through lidity of (9) and a derivation of the form of US (R) in (11) is given in the appendix, starting from a formally exact ∂ dRi ∂ d “entangled” wavefunction Ψ(x, R)= n χn(R)ψn(x, R) Ci = χ = χ . (19) ∂Ri dt ∂Ri dt for the complete object composed ofP system and envi- Xi Xi ronment. It will be shown there that the asymmetry condition (10) justifies a posteriori the form (9) of the Since the Ri are reduced to classical variables Ri(t) (14) wavefunction. As is well known from adiabatic approxi- can now be written mations in other contexts a wavefunction of the form (9) ∂ (H + H − i¯h − U (t))ψ(x, t)=0 (20) can only be justified a posteriori if a condition such as S ES ∂t S x (10) is fulfilled. Backed by the asymmetry condition (10) and (11) we Here we emphasize that the time derivative is to be taken determine χ from the eigenvalue equation with {x} fixed since it arises from the derivative w.r.t. the independent (quantum) variables {R}. Finally, a phase (HE + US (R) − E)χ(R)=0. (12) or gauge transformation

The term US represents the very small inﬂuence S has on t ′ ′ ˜ the state of the environment. The environment is taken ψ = exp[i/¯h (US (t ))dt ]ψ (21) Z to be a large, quasiclassical system so that its state vector −∞ χ can be approximated by a semiclassical wavefunction leads to the TDSE for the quantum system alone. To write it in a familiar form we might specify, although not χ(R)= A(R) exp(iW/¯h), (13) necessary, HES = V (x, t), i.e. the interaction with the environment is expressed as a potential. Then we have where W (R, E) is the (time-independent) action of the from (20) and (21) classical Hamiltonian HE . Inserting (7) and (9) into (1) we get using (12) ∂ (H + V (x, t) − i¯h )ψ˜(x, t)=0. (22) S ∂t

3 Having accepted (22), the whole structure of time- quantum system alone, i.e. where the operators are those dependent quantum mechanics, e.g., the transition to of the quantum system. In that case HES = 0 and the Heisenberg and interaction pictures, time-dependent per- quantum systems satisfy the TISE (6). A time energy turbation theory etc., can be developed as usual. Indeed, ‘uncertainty relation” then only arises by introduction of Briggs and Macek show explicitly that, in the same ap- a mathematical time through Fourier transform of (6) to proximations that lead to (22), the time-independent T- a “time” space. The uncertainties ∆E and ∆t then refer matrix element for the system plus environment reduces to the widths of Fourier distributions and the uncertainty to a time-dependent transition amplitude for the system relation to the inverse relation between them. alone. Similarly, a precise consideration of the nature of the interaction V (x, t) should allow one to derive the many variations of stochastic Schrödinger equations that III. SPECIFIC EXAMPLES have been proposed to model the interaction of a quantum system with an environment or measuring device. To illustrate the general approach of section II in more One further observation must be made. This is the detail we will consider three examples which represent question of the “uncertainty relation” for energy and the most common ways in which quantum systems are time. Since there is no canonical operator for time there probed and measured. First we will discuss the interac- is no uncertainty relation in the sense of Heisenberg. tion of the quantum system with a particle beam, then That quoted in many books arises from the basic prop- we will describe the interaction with a ”bath” of oscil- erty of the Fourier transform from energy space to time lators, e.g. photons or phonons. Finally, we will show space in which time appears as a mathematical, rather that also in a relativistic environment the time-dependent than a mechanical (physical) variable. However, within Dirac equation (TDDE) can be derived from the time- the approximation of the environment as a classical ob- independent Dirac equation (TIDE) in a way analogous ject a time-energy relation for the quantum system can to the non-relativistic case. be derived from the uncertainty relation for the environment, since it is the position variable of the environment that defines the classical time. For any two operators A. A particle beam as environment related to the environment, one has ~ ~ 1 A particle beam of fixed momentum P =h ¯K inter- ∆A∆B ≥ h[A, B]iR. (23) acting with a quantum system was considered by Briggs 2i and Macek [4]. The asymmetry condition (10), neces- In particular if A = HE and B = Ri, then sary to separate environment and quantum system, is achieved when the beam kinetic energy P 2/2M is much hPii greater than than the energy differences ∆E in the sys- ∆HE ∆Ri ≥ ¯h/2 . (24) S M tem states populated as a result of the interaction. The Now, in the classical limit for the environment variables, (semi-)classical limit for the environment, necessary to ~ ∆Ri = vi∆t and vi = Pi/M = hPii/M, so that we obtain justify (13), is reached when P is so large that the de Broglie wavelength is far shorter than the extent of the ∆EE ∆t ≥ ¯h/2. (25) quantum system. In this case the WKB-wavefunction for a free beam-particle is the exact quantum solution. We However, from (12) and (11) E = EE + ES where E is may choose a coordinate system with the z−axis along the fixed total energy. Hence, ∆EE = ∆ES and (25) the beam direction, becomes −3/2 χ(X,Y,Z)=(2π) exp(iPZ Z/¯h), (27) ∆ES ∆t ≥ ¯h/2, (26) where P~ = (0, 0, PZ ) is the classical momentum of the i.e. the energy-time uncertainty for the quantum system beam. Since (27) is of the form (13) it leads directly to emerges from the fluctuations in the expectation values the emergence of time according to (19), of the environment variables. Note that for the derivation and application of this un- P ∂ dZ ∂ d Z = = . (28) certainty relation it is necessary that the quantum sys- M ∂Z dt ∂Z dt tem interacts with the environment through the potential HES . It is this interaction that leads to the uncertainty in the system energy ES . In the same way the uncertainty in the time ∆t arises from the time development B. A quantized field as environment described by (22) where the time is the classical time defined by the classical time-development of the envi- In the second example the environment comprises a ronment. Such energy-time relationships are to be dis- collection of quantum oscillators with mode frequencies tinguished from those usually postulated for the isolated

4 ωk, i.e. photons or phonons. Then the Hamiltonian HE but is now a product of a spinor χ(R~) representing the can be written as a sum over field modes spin state of E and a spinor ψ(~x, R~) representing the spin 2 2 2 state of the system but depending parametrically on the Pk ωkQk HE = + , (29) space variables of the environment. 2mk 2 Xk The analogue of (14) now becomes where mk = 1 for photons. The field operators satisfy [χ(EE − E + HS + HES ) − c~αE χP~E ]ψ(~x, R~)=0. (35) the usual commutation relations However, the operator c~αE is just the velocity operator [8] [Qk, Pk′ ]= i¯hδkk′ (30) 2 which for positive energy solutions has the form c P~E /EE . For free motion the exact solution is the same as the with semiclassical one. However Jensen and Bernstein [9] have ∂ shown that even when potentials as in (34a) and (34b) Pk = −i¯h . (31) ∂Qk are retained, in lowest order semiclassical approximation the form of the velocity operator is unchanged. Then, in 2 The energy EE is the eigenenergy of (29) and χ(Q) is an this limit, with EE = Mc one has eigenvector in the Q-representation which need not be specified further here. Then, using (9), we write dR~ P~ /M = ~v = (36) E E dt Ψ(x, Q)= χ(Q)ψ(x, Q). (32) so that (35) becomes Substitution of Ψ into (1) leads as before to (14). The quasiclassical field limit leads now for the operator Ci in [HS + HES − i¯h~vE ∇R]ψ(~x, R~(t)) = US ψ(~x, R~(t)) (37) (14) to the replacement or Pk dQk Ci = χ → χ (33) ∂ mk dt H + HES − i¯h ψ(~x, t)= US (t)ψ(~x, t). (38) S ∂t x where Qk(t) is now a classical field amplitude. Since (33) is of the same form as (18) time emerges as in (19) and With the phase transformation (21) one has the TDSE is established for the case of a quasi-classical field as environment. ∂ H + HES − i¯h ψ˜(~x, t)=0, (39) S ∂t

C. An example of relativistic dynamics the time dependent Dirac equation. Here, it is interesting to note that the time coordinate of the quantum system Finally, we consider how the relativistic generaliza- spacetime arises from the space coordinate of the classical environment tion of the transition from the TISE to TDSE occurs . for femions, i.e. how a time-independent Dirac equation (TIDE) for system plus environment becomes a time- IV. SUMMARY dependent Dirac equation for the system. To keep the derivation simple we restrict ourselves to a quantum object of two fermions whose spins are uncoupled and where We began our considerations with a time independent the energies are such that pair production can be ne- stationary state of a complete object comprising system glected. In this case the classical limit will be where the and environment. The semiclassical treatment of the en- relativistic mass M of the environment fermion becomes vironment E with the requirement that its own state and much greater than that of the system fermion m. The energy to zeroth order are unaﬀected by the quantum TIDE can be written formally as in (1) with the Hamil- system S, has led to a TDSE for this system in which tonian (7) whose elements are now deﬁned as, the quantum variables of the environment are replaced by classical variables. The interaction with the envi- 2 HE = c~αE P~E + VE (R~)+ βE Mc (34a) ronment then appears as explicitly time-dependent and ~ 2 the motion of the environment provides a time derivative HS = c~αS PS + VS (~x)+ βS mc . (34b) which monitors the development of the quantum system. If the interaction with the environment is ignored i.e. The potentials V , V are potentials acting separately E S the quantum system is closed, then time is reduced to on environment and system particles, respectively and a mere mathematical variable and can be removed en- can be neglected in what follows, i.e. we consider two tirely by the simple phase transformation (5) leading to free fermions interacting through a coupling Hamiltonian the TISE of (6). Note, however, that it is inconsistent HES . The total wavefunction Ψ is then written as in (9)

5 2 2 2 to put V (x, t) = H in (22) to zero in order to ob- ¯h ∂ ∂ ¯h ∂ ES − hψ | |ψ i + hψ | |ψ i χ tain a TDSE of the form of (3) which is simply postu- m 2M ∂R2 n m ∂R n 2M ∂R n Xn,i i i i lated. Were HES zero, the TDSE (22) cannot be derived, since the Hamiltonian (7) is then fully separable in x and = Eχm (A3) R and instead of the approximation (9) one has an ex- This is the full quantum equation for the environment act solution of the form Ψ(x, R) = χ(R)ψ(x). This has whose states χ are mixed by the ”back-coupling” from the consequence that system and environment are fully n the system. The first set of coupling terms on the lhs. decoupled implying that the environment can no longer of (A3) are called potential couplings and usually the provide time for the system. Formally, one can see this ψ are chosen to diagonalize these terms. The remain- from the uncertainty relation (26). As H → 0, the en- n ES ing terms are the ”dynamical couplings”, since their off- ergy exchange between environment and system vanishes diagonal matrix elements describe changes in the state of and E and S separately become isolated without uncer- the quantum system induced by the motion of the envi- tainty in in their respective energy. Hence, in (26) with ronment. Clearly, in order to fulfill the conditions that ∆E → 0 we get ∆t → ∞. In this sense time arises and S we demand for separation of environment and system, is meaningful for a quantum system only when interac- it is necessary that all off-diagonal couplings are small, tion with a quasi-classical external environment defines i.e. the environment is insensitive to changes in the state a clock (e.g. an oscillator) with which the time develop- of the system. Then (A3) reduces to the single-channel ment is monitored. equation

2 ∂ ¯h ∂χ ACKNOWLEDGMENT (H + E (R) − E)χ (R)= hψ | |ψ i m E m m m ∂R m 2M ∂R Xi i i We would like to thank L. Diosi for helpful discussions. (A4)

where APPENDIX A: 2 ¯h ∂2 E (R)= hψ |H + H − |ψ i. (A5) m m S ES 2M ∂R2 m In the following we will show under what approxima- Xi i tions and with which consequences an exact or “entan- The dynamical coupling involving the second derivatives gled” wavefunction, 2 ∂ 2 has been incorporated formally in Em(R). However, ∂Ri Ψ(x, R)= χn(R)ψn(x, R) (A1) in connection with the semiclassical approximation for Xn χm which must be made to derive the TDSE for the system, it is consistent to neglect this term. This is shown for the large object described by the Hamiltonian H in explicitly in section II in the reduction of the operator Ci. (1), leads to the product wavefunction (9). This form of Note that this forces a choice of the environment such the wavefunction describes a state of H = HE +HS +HES that the major R-dependence is contained in the χm and if the system S and the environment E are weakly cou- the ψ are slowly varying functions of R. Then, since pled under the asymmetry condition (10). Without loss m 2 of generality we may assume the ψn to be orthonormal ∂ ∂ 2 for each R, i.e. hψn|ψmi = δnm, where here and in the hψm| 2 |ψmi = |hψm| |ψni| , (A6) ∂R ∂Ri following brackets h|i denote integration over system vari- i Xn ables x only. the requirement that the term on the lhs. of this equation We first proceed exactly as in the Born-Oppenheimer, is small ensures that the off-diagonal dynamical couplings or better, perturbed stationary states (PSS) approxima- in (A3) are also small. tion of molecular physics [5] where (A1) is substituted in If the ψm can be chosen real, the dynamical coupling HE + HS + HES − E|Ψi = 0 and a projection is made terms on the rhs of (A4) vanish. If the ψm are complex, onto a particular state ψm to give these terms give rise only to geometric (or Berry) phases that can be accounted for by a phase transformation of hψ |HE + HS + HES − E|ψ iχ = 0 (A2) m n n the χn. Effectively, then (A4) reduces to the eigenvalue Xn equation

Making use of the explicit form of HE given in (8) we (HE + Em(R) − E)χm(R) = 0 (A7) see that (A2) describes a state χm of the environment ’closely coupled’ to all other states χn: for the state of the environment when the quantum system is in the state ψm. Note that the environment is still HE χm + hψm|HS + HES |ψniχn coupled to the system in that the diﬀerent states of the n X system provide separate potential surfaces Em(R) for the

6 motion of the environment. The complete independence of the environment from the precise state of the system is achieved in the approximation that the diﬀerences in the Em(R) can be replaced by an average potential leading to a common R dependence for all χm(R), i.e., χm = amχ, where the am are constants. This gives the simpliﬁed form of (A1)

Ψ(x, R)= χ(R) anψn(x, R) ≡ χ(R)ψ(x, R) (A8) Xn corresponding to the ansatz of (9). Similarly (A7) becomes identical to (12), where the averaged potential US of (11) assumes the form

2 US (R)= hψ|H − VE (R)|ψi = |am| Em(R) (A9) Xm with Em(R) from (A5). Note that the product ansatz (A8) ((9) of the text) and the environment equation (12) evaluated in the lowest order WKB approximation lead directly to the TDSE (22) for the quantum system. The analysis of this appendix shows how the environment must be chosen ”large enough” so that it is insensitive to the back-coupling from the system. This insensitivity is necessary to derive an eﬀective TDSE from the TISE for the composite object of system coupled to environment.

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