Postexponential decay of the probability density of a quantum particle leaving a trap can be reproduced accurately, except for interference oscillations at the transition to the postexponential regime, by means of an ensemble of classical particles emitted with constant probability per unit time and the same half-life as the quantum system. The energy distribution of the ensemble is chosen to be identical to the quantum distribution, and the classical point source is located at the scattering length of the corresponding quantum system. A one-dimensional example is provided to illustrate the general argument.
Postexponential decay of the probability density of a quantum particle leaving a trap can be reproduced accurately, except for interference oscillations at the transition to the postexponential regime, by means of an ensemble of classical particles emitted with constant probability per unit time and the same half-life as the quantum system. The energy distribution of the ensemble is chosen to be identical to the quantum distribution, and the classical point source is located at the scattering length of the corresponding quantum system. A one-dimensional example is provided to illustrate the general argument.
We study the elusive transition from exponential to postexponential (algebraic) decay of the probability density of a quantum particle emitted by an exponentially decaying source in one dimension. The main finding is that the probability density at the transition time, and thus its observability, increases with the distance of the detector from the source up to a critical distance beyond which exponential decay is no longer observed. Solvable models provide explicit expressions for the dependence of the transition on resonance and observational parameters, facilitating the choice of optimal conditions.
We derive an analytical expression for the propagation of a quantum particle at asymptotic long distances and times from the potential where the particle was initially confined. We obtain that the particle is described by an evolving resonance in time domain that possesses a peaked shape characterized by the resonance parameters of the dominant resonance coefficient of the decaying wave function. The above situation corresponds to an unexplored postexponential regime that represents the ultimate fate of a decaying particle.
In (1) it is shown how the Schrodinger equation can be “forced” in a sense to follow from the Fokker-Planck classical stochastic equation which contains a classical probability density. Other approaches in the literature which are similar use continuity equations and also a classical density. The key to all of these approaches is to assume, in an ad hoc manner it seems, that classical probability is equivalent to W*(x)W(x) where W(x) is a complex function called the wavefunction. In principle, one may write a density as W*(x)W(x). This is mathematically correct even if, for the sake of argument, it is ad hoc. The problem is that for a constant density (i.e. constant momentum and speed), one does not know if W(x)=exp(ipx) or W(x)=exp(-ipx). We consider here the arguments shown in (1). If a free particle has a density(x)=1/L, it may be associated with W(x) = exp(ipx)/sqrt(L). A free particle density, however, is associated with constant p (momentum) and assumes one knows x(t). If one considers the free particle to be trapped in a one-dimensional box with infinite potential walls, it travels to the right with p and then to the left with -p. The density(x) classically is 1/L, where L is the length of the box. The density is constant in time, but the direction of motion is a function of time. Thus, information is lost considering an equation such as the Fokker-Planck equation which uses a density function which does not track the direction of motion in time. In particular, writing density(x) = W*(x)W(x), which is done in (1) and applying this density to a bound particle in a box with infinite walls leads to an equation linear in W(x). Thus exp(ipx) and exp(-ipx) are both possible solutions if V(x)=0. Given that the conservation type equation obtained from the Fokker-Planck (or continuity) equation is linear in W, exp(ipx)+exp(-ipx) is a solution. Thus there is interference which leads to a density which does not match a constant classical density which should apply to a particle moving to the right and then to the left in a bound state because one assumed a classical picture a priori. Simply suggesting that density classical = W*(x)W(x) mathematically does not mean that the a priori classical picture may be dropped. As a result, we argue that there exists a contradiction in approaches which write classical density as W*(x)W(x) and then obtain the Schrodinger equation. As a remedy, we suggest that W(x) does not follow from a theory of classical density=W*(x)W(x), but vice versa. One needs equation which deal with W(x) a priori, and once it is found, one may create a classical density through W*(x)W(x).
Classical mechanics traces a particle’s movement through space and time through x(t). For example, one may consider the case of a particle moving at constant speed (momentum p) in one direction or a particle bouncing back and forth at constant speed in a box. These problems are described in time and have direction and position associated with time etc. It is possible to define a probability P(x,t) in the classical picture based on delta time, the amount of time a particle spends in a little region of space dx. P(x,t), by itself, loses much relevant information present in the x(t) scenario because a particle moving to the right, or to the left, or bouncing back and forth all have the same probability, 1/L, where L is length. Furthermore, this probability is the same for all p (momentum) values. Thus a probability description, without extra information, is a subset of the relevant information in the problem. As a result, to describe the classical picture fully, one might say that a particle has a probability of 1/L, but also a mass mo and velocity +v (i.e. momentum mov nonrelativistically). For the particle bouncing back and forth in a box with constant p, one needs to give the initial position at time=0 and then a subsequent time (as well as the initial velocity and mo) in order to fully describe the problem, even though probability = 1/L. In the literature (see (1)), there have been many attempts to derive the quantum Schrodinger equation from a classical Fockker-Planck equation or a continuity equation (which follows from the Fokker-Planck equation as shown in (1)). The continuity equation contains a probability P(x,t) and a velocity(x,t) and is generally argued to represent classical physics. It does, if one retains appropriate classical conditions such as the discontinuity of density for a particle in a box (i.e. density=0 outside the box, but 1/L inside) and the notion of differing directions within the box. In fact, one may write density=W*(x,t)W(x,t) where W(x,t) is an ad hoc complex function and still retain compatibility with classical physics. If one does not, however, retain the classical conditions completely, nonclassical physics is introduced, even if density(x,t)=1/L. For example, in quantum mechanics density(x) is considered to be continuous in all x. Thus for a particle moving in a box, density outside the box is 0 so at the box wall it must also be 0. In the classical case, the density is 1 inside the box and 0 outside. There is a discontinuity classically and this cannot be changed and this contradicts the quantum mechanical picture. Thus quantum mechanics does not follow form classical Fokker-Planck or continuity equations which retain classical conditions.. Furthermore, because time is completely removed in the FP-continuity approach as applied to quantum mechanics, one cannot distinguish between forwards and backwards motion. As seen by the velocity field given in (1), v is zero for a bound particle which completely removes the sense of time and violates the classical physics conditions. It is, however, not zero for a particle moving in one direction. Thus there seems to be no issue in describing a Fokker-Planck-continuity equation as describing classical physics. One may even introduce density(x,t)= W*(x,t)W(x,t) provided one retains classical conditions allowing for discontinuity in density and different W(x,t) values at different times i.e. a discontinuity in W(x,t) at a box wall. If these conditions, however, are not met, then one no longer describes classical physics, but a new kind of physics (quantum physics). Thus one cannot argue that quantum mechanics follows from classical statistical Fokker-Planck-continuity equations. It may follow from these if one adds a set of extra conditions. Given these extra conditions, however, one could derive quantum mechanics without starting with classical density.
In (1) the Wigner transform of an oscillator is calculated and the high n energy level limit taken using the WKB approximation yielding a result proportional to delta(p - p(x)) + delta(p+p(x)) where p(x) = sqrt(2m(E-.5kxx)). The Wigner transform involves quantum wavefunctions which for bound systems include +/- p components which interfere i.e. a(p)(exp(ipx) + exp(-ipx) for a(p)=a(-p). Classically, a bound particle moves to the right during one time interval and to the left during the other so there is no interference of +/-p. The quantum wavefunction retains +/-p interference for all energy levels and so does not portray the full physical information of the bound system i.e. leads to the two delta functions. For high n (energy level), crests and troughs are very narrow compared to the length of the system, approximating dx units of length. Thus, uncertainty in time is only within these tiny crests/troughs, otherwise there is a distinct time position relationship as in classical physics, but the wavefunction W(x) does not show it. How then may the wavefunction be interpreted? We argue the time independent wavefunction, say for a bound state, represents an averaged steady stream picture which is often not the physical picture. For example, a bound particle is not a steady stream average nor is a finite number of photons hitting an interface of media with different indices of refraction/reflection. Nevertheless, the steady stream approach with time removed may be used to solve the problem i.e. ultimately find a one dimensional spatial density (which averages out the original two dimensional probability of exp(ipx), energy levels etc. For instance, the energy levels of a bound system do not depend on following the particle in time. We raise this issue because we think it may have implications on the calculation of entropy in bound single particle quantum systems. In general, a spatial Shannon’s entropy is defined as: - W(x)W(x) ln[W(x)W(x)] (one dimensional problem). W(x) (wavefunction) is real because p/-p pieces interfere in a steady stream picture. A single bound particle, however, is not a steady stream picture. exp(ipx) is a dynamic probability equal to cos(px)+i sin(px) to indicate the direction of motion. Simply because exp(ipx)+exp(-px) removes the complex portion (i.e. for a steady state picture with a(p)=a(-p)) does not mean that this represents the full physical picture especially as energy levels become high and crests/troughs become narrower with respect to the overall system length it seems there is no justification for the steady stream interference picture.. In other words, using a real W(x) in a Shannon’s entropy expression treats one dimensional probability W(x) = bound wavefunction, but this is not the full probability in the picture, we argue. Full probability is two dimensional and demonstrates direction for many energy levels (especially high ones). For a free particle exp(ipx), cos(px) and sin(px) are essentially the same distribution. Thus cos(px)cos(px) describes the probability distribution in x. For W1(x)=Sum p>0 a(p)exp(ipx), Real(W1(x)) is not equal to Imaginary(W1(x)) because one does not have a simple shift of the real distribution to obtain the imaginary one. The real distribution is chosen to represent probability because the interference of exp(ipx), exp(-ipx) terms removes the imaginary piece, but it is removed only on average. If physically exp(ipx) and exp(-ipx) should not interfere, then one may need to take some kind of average of Real(W1(x))Real(W1(x)) and Imaginary(W1(x))Imaginary(W1(x)) and use this in Shannon’s entropy.