Schrodinger Equation from Fokker-Planck Equation and Contradiction Due To Interference? Part II
Francesco R. Ruggeri
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.
•Apr 24, 2023•0 citations