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Physical Review A

Quantum theory of particle motion in a rapidly oscillating field

Richard J. Cook, Donn G. Shankland, Ann Laurie Wells

Published Feb 1, 1985• 160 citations
Publisher

Abstract

We show that the motion of a particle of mass m in a high-frequency time-dependent potential V(x\ensuremath{\rightarrow})=v(x\ensuremath{\rightarrow})cos(\ensuremath{\Omega}t) is governed by a Schr\"odinger equation with time-independent effective potential ${V}_{\mathrm{eff}}$(x\ensuremath{\rightarrow})=\ensuremath{\nabla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow})\ensuremath{\cdot}\ensuremath{\nabla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow}) /4m${\ensuremath{\Omega}}^{2}$. The validity of this approximation and an exact formal solution based on the Wigner function are discussed for the case in which v(x\ensuremath{\rightarrow}) is quadratic.

DOI
10.1103/physreva.31.564

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Nov 13, 2000•3 citations
We show that the motion of a particle of mass m in a high-frequency time-dependent potential V(x\ensuremath{\rightarrow})=v(x\ensuremath{\rightarrow})cos(\ensuremath{\Omega}t) is governed by a Schr"odinger equation with time-independent effective potential Veff{V}_{\mathrm{eff}}Veff​(x\ensuremath{\rightarrow})=\ensuremath{\nabla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow})\ensuremath{\cdot}\ensuremath{\nabla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow}) /4m\ensuremathΩ2{\ensuremath{\Omega}}^{2}\ensuremathΩ2. The validity of this approximation and an exact formal solution based on the Wigner function are discussed for the case in which v(x\ensuremath{\rightarrow}) is quadratic.
The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.
Addendum Feb. 4, 2021 In this note, a potential of .5kxx + f(t)x is considered together with a product wavefunction Wa(y,t)exp(i y db/dt) where y=x-b(t). Product wavefunctions when inserted into the Schrodinger equation lead to two equations. The first may be chosen to depend only on one product e.g. Wa. In this case, it is a harmonic oscillator Schrodinger equation with an extra c(t)Wa term which may be handled by a pure time phase i.e. exp(i Integral (0,t) c(t1)dt1)). The second equation incorporates the potentail f(t)x and involves a coupling term -1/m dWa/dy d/dy exp(i y db/dt). This term, however, is exactly canceled by a term from i d/dt (partial) Wa(y-b(t)) exp(i y db/dt) so the coupling is eliminated. Thus, one has a second differential equation in b(t) linked to the potential f(t)x or f(t)y- f(t)b(t) which is completely uncoupled. Thus, there are two decoupled types of motion. Harmonic motion in y=x-b(t) and collective motion related to b(t) linked directly to f(t). Addendum Feb. 3, 2021 The term -1/2m d/dy d/dy W1 in ((6)) is db/dt db/dt (1/2m) and is incorporated in c(t)Wo in equation ((5)). Also i d/dt (partial) {exp(i(x-b/m)db/dt) W(x-b/m)) = -i/m dW/dy db/dt +(1/m)db/dtdb/dt -yd/dt db/dt as seen in the equation directly below ((6)) on the LHS. An extra pure time piece is added i.e. +(1/m)db/dtdb/dt. The “time-independent” Schrodinger equation appears to be directly related to stochastic motion, we argue. If there is no potential, the wavefunction exp(ipx) is statistical, indicating periodic motion with a norm of one for all x. If a time-independent potential is added, it may be decomposed as V(x)=Sum over k Vk exp(ikx). Thus, time is present not just in exp(-iEt), but in the stochastic behaviour of the potential which is only V(x) on average. At each time, one does not know which Vk acts. Thus, the time-independent Schrodinger equation could perhaps be called the purely stochastic Schrodinger equation. It is possible to have a time dependent Schrodinger equation with a time dependent potential. Sometimes the time in the potential is hidden by a variable change as the equation involves partial derivatives. We try to argue that this potential contributes to collective motion. Time is already present in V(x), but in a stochastic manner. Explicit time (or a time related variable change) means a lack of stochasticity as one follows the potential in a predictable manner in time. It is possible that collective motion may be associated with a reference frame. For example, one may remove center-of-mass motion and consider only internal motion i.e. the quantum stochastic internal behaviour. In this note, we briefly examine two examples. The first is a free quantum particle viewed from a constantly accelerating frame as discussed in (1). In such a frame, the free particle appears as a particle in a gravitational potential +mgx1, but x1= x-X(t) so X(t) suggests collective motion. One may argue that x represents stochastic motion, but this is not necessarily the case because one may have collective potential energy linked to collective kinetic energy. In other words, id/dt partial W may yield a term which cancels mgx indicating that it does not act as Sum Vk exp(ikx). The second example involves a quantum oscillator with an extra xf(t) potential. We argue that this leads to collective motion as the problem may be recast in terms of a variable y=x- b(t) and solved as a “time-independent oscillator” in y as discussed in (2). Again, there exists the notion of a kind of collective motion or extra frame which leads to a phase factor exp(iy d/dt partial b(t)) as the stochastic quantum behaviour defines the physical density and the phase term exp(i phase) disappears from the density). In such a case, W(y) the time-independent oscillator solution may be written as: Sum over p a(p) exp(ipy), but y contains t so the statistical nature of exp(ipy) is somewhat altered frome exp(ipx) i.e. there is an extra time related phase exp(i p f(t)) which has the appearance of a kind of collective motion as all p values are in synch with the same f(t) which may be sin(wt) for example.
In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).
We investigate the many-body wave function of a quantum system with time-dependent effective mass, confined by a harmonic potential with time-dependent frequency, and perturbed by a time-dependent spatially homogeneous electric field. It is found that the wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. The wave function reduces to that of the harmonic potential theorem wave function when both the effective mass and frequency are static. An example of application is also given.
We present a general procedure for obtaining equations of motion for the Wigner distribution of functions that are governed by ordinary and partial differential equations. For the case of fields we show that in general one must consider Wigner distribution of the four variables, position, momentum, time and frequency. We also show that in general one cannot write an equation of motion for position and momentum however it can be done in some cases, the Schrodinger equation being one such case. Our method leads to an equation of motion for the Schrodinger equation with time dependent potentials in contrast to the result obtained by Wigner and Moyal which was for time independent potentials.