Steady state scenarios exist in classical physics and even though there is motion (flows) (i.e. changes in time) overall the system does not change in time. Such systems are usually not described in terms of probability. Furthermore, a single particle moving in one direction is described by x(t), i.e. x=vt, so there is no probability and no steady stream scenario.Such a particle, however, may give rise to a steady stream scenario if it encounters a V1-V2 discontinuity (V1, V2 are two constant potentials) and E (energy) remains constant. In such a case, there is a probability to reflect or to pass into the V2 region with a different p, but the same E. This situation automatically introduces probability and may be associated with a steady stream picture, even in classical physics. As a result, classical physics is not as deterministic as it may seem. If probabilities arise at the discontinuity, one may ask: Is there a steady-stream (flux) type of probability associated with a moving particle? In a steady stream situation there is no change in time, but a moving particle clearly changes in time x(t). How does one associate probabilities which do not contain time to flow problems which clearly contain motion? We investigate these questions and try to argue that the quantum exp(ipx) form and the Born W*(x)W(x) wavefunction picture are associated with these questions.
Steady state scenarios exist in classical physics and even though there is motion (flows) (i.e. changes in time) overall the system does not change in time. Such systems are usually not described in terms of probability. Furthermore, a single particle moving in one direction is described by x(t), i.e. x=vt, so there is no probability and no steady stream scenario.Such a particle, however, may give rise to a steady stream scenario if it encounters a V1-V2 discontinuity (V1, V2 are two constant potentials) and E (energy) remains constant. In such a case, there is a probability to reflect or to pass into the V2 region with a different p, but the same E. This situation automatically introduces probability and may be associated with a steady stream picture, even in classical physics. As a result, classical physics is not as deterministic as it may seem. If probabilities arise at the discontinuity, one may ask: Is there a steady-stream (flux) type of probability associated with a moving particle? In a steady stream situation there is no change in time, but a moving particle clearly changes in time x(t). How does one associate probabilities which do not contain time to flow problems which clearly contain motion? We investigate these questions and try to argue that the quantum exp(ipx) form and the Born W*(x)W(x) wavefunction picture are associated with these questions.
Classical mechanics usually employs two variables x,t to describe motion in the form of x(t). Derivatives of this function yield velocity, acceleration etc. For a bound system, velocity (momentum) changes from one x to the next, but one does not necessarily need time to describe the scenario as one could use the pair of variables p(x),x instead of x(t),t. An issue which arises is that p(x) has double values as it is positive when the bound particle is moving to the right and negative when it is moving to the left. In such a case, time distinguishes between the two motions which occur at the same spatial point. From the point of view of energy, however, such a degeneracy at x is irrelevant because kinetic energy is pp/2m. Thus, it seems one may remove time from a bound state problem as long as one uses an energy (i.e. energy conservation) picture. As a result, we propose using a p(xi), xi model to describe a bound state with p(xi) (momentum at xi) being different at each xi. In order to calculate average of the absolute value of p at xo we suggest using a Dirac delta function delta(p-p(xo)) i.e. related to {Integral dp p delta(p-p(xo))} . If one expands the delta function in terms of the basis exp(ipx), it seems that quantum mechanics for the bound state arises as a classical steady stream picture in terms of energy conservation, with exp(ipx) representing a steady stream picture of a particle with a constant momentum. We discuss informational probability properties of exp(ipx) which make it the basis of choice.
We try to argue that the idea of probability associated with motion may have classical beginnings in that the center-of-mass (cm) of a piece of mass moving with constant velocity v is present within a dx at some time, but moves out of dx. Flux is constant, but at one time the particle cm leaves dx and arrives at the adjacent dx. Constant flux suggests one does not need to consider time. In such a case, one might associate a periodic function showing probability being removed from a region dx (at x) (i.e. a negative distribution) followed by it being added (a positive distribution) to a region x+dx. This suggests a negative-positive cycle, but this labels certain x regions as negative and others as positive which is inconsistent with all x regions having the same weight based on a constant v. Given that time is removed, it seems this cycling may have nothing to do with velocity v or perhaps anything else for that matter. As a result, one may postulate an Aexp(ibx), where b is a constant, for motion. We will argue for a phase linear in x. AAexp(-ibx)exp(ipx) yields a constant AA, which is flux, and treats all x points the same. At this point, one cannot go farther unless one considers other issues.We consider two such issues. First we ask what happens if the particle is viewed from a frame moving at constant v? The velocity of the particle will not be the same, but we have argued that exp(ibx’) may be the form, yielding no information about b when (x,t) changes to (x’,t’). If on the other hand, the phase cannot change if the particle is viewed from the moving frame, one requires a Lorentz invariant. This suggests that the phase may actually be -Et+px. (We also show that it is not necessary to use special relativity to obtain the form exp(ipx). Rather one may consider ideas of pressure balance associated with continuity of exp(ibx).) As a second consideration, we examine a one dimensional reflection-refraction problem at a point xo for which a probability equation 1= Probability(reflect)+Probability(refract) exists. If exp(ipx) for constant E describes motion, this motion changes with changing p, yet one wishes probability to be a continuous function at xo. This suggests different weight coefficients for different exp(ipx) which seem to be associated with P(reflect), and P(refract). If x=0, one has an equation in the weights, with A+B (x<xo) matching C (x>xo) at xo. If one stops at this point, exp(ipx) does not seem to have any impact. One may note, however, that in general xo is not zero and that one wishes at the very least to have the derivatives match. This immediately links the A,B,C values to the p (momentum) discontinuity. Thus using special relativity to create an exp(ipx) has practical implications because it allows one to deal with p discontinuities. We also show though that special relativity, i.e the use of -Et+px, may be bypassed and the form Aexp(ipx) obtained as a result of its first derivative yielding p which is needed in a pressure balance equation. This then suggests A is the square root of flux.
In (1) it is argued that a quantum particle’s probability P(x,t) satisfies a Nelson stochastic equation: d/dt partial P(x,t) + d/dx partial (b(x,t)P(x,t)) = D d/dx d/dx partial P. ((1)) where b(x,t) = hbar/m d/dx partial S(x,t) + 2D d/dx partial ln(R(x,t)) ((2)) with W(x,t)=wavefunction= R(x,t) exp(i S(x,t). Thus both a quantum wavefunction (satisfying the Schrodinger equation) and a separate P(x,t) are present. If P(x,0)= W*(x,0)W(x,0) ((3)), i.e. Born’s rule, then this rule holds for later times. It is, however, suggested in (1) that if P(x,0) not= W*(x,0)W(x,0) then the system may relax over time into a situation for which ((3)) holds. In this note, we argue that for a free particle moving with constant p in one direction, P(x,t)=constant, but there is an underlying two dimensional wavefunction exp(ipx). We ascribe this to two x-probability distributions associated with motion. In particular: v= (xfinal - xintial) / delta t. If xfinal and xinitial are both distributions such that on average v = constant, one needs to give a probability for both i.e. have two dimensions or an ordered pair. The two probability distributions, however, are not independent as they must describe a constant velocity i.e all points of space being the same. This leads to exp(ipx) such that exp(-ipx) exp(ipx) =1 . In other words, one may constrain two probability distributions to yield a deterministic type result for a single particle which appears as a probabilistic result for a steady stream.Born’s Rule automatically applies to a free particle and is not a “postulate” as argued in (1). The form exp(ipx) follows from the Lorentz invariant -Et+px we argue. We further argue that if one creates a linear superposition of exp(ipx)s i.e. W(x)=Sum over p a(p)exp(ipx) then Integral dx W*(x)W(x) is always linked to a steady stream probability picture ie. Sum over p a*(p)a(p) = 1. This is why given a p, one does not know its x point- it is associated with a steady stream. As a result, we argue that there is always a meaning for Born’s rule W*(x,t)W(x,t). As a result P(x,t=0) not= W*(x,t=0)W(x,t=0) must describe a probability which differs from the steady stream scenario linked with exp(ikpx).
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.
One may consider constant fluxes which exist for all x points in space. These are associated with a constant velocity and a momentum. Consider next a discontinuity which does not change energy. For example, one may have energy E, momentum p and a constant potential V1 which suddenly changes to V2 at x=xo with E remaining constant. This leads to a momentum discontinuity as well as a flux discontinuity, but no work is done. As a result, there exists a pressure balance (related to an incident flux, a reflected one and the new p2 flux for x>xo), i.e. there is no discontinuity in pressure. A single constant flux also has no discontinuity in pressure either. We suggest there should exist a continuous function of momentum and x which allows for/ensures this pressure continuity. The momentum discontinuity may be at any xo, but the pressure balance is the same.As a result, one would expect this function and its first derivative to be continuous at xo, but a combination of the two should eliminate the function (of p and xo) leaving only a pressure balance equation. In other words, two equations need to combine to create a single classical pressure balance equation, suggesting that there may exist a square root flux type probability function. We suggest that pressure balance at momentum discontinuity points (with E remaining constant) may be the only consideration necessary to ultimately obtain the quantum free particle exp(ipx). An advantage of this pressure-only viewpoint is that it accounts for weighted averages of exp(ipx) having the same form, namely exp(ipave(rms) x) and exp(-ipave(rms) x) in a potential with infinite walls. Pressure considerations may be applied to a single particle exp(ipx) or an ensemble, yielding the same form if the same conditions apply. We argue that both a square well potential and infinite well potential are examples of momentum discontinuities, except in this case one has discontinuities in ensemble values (i.e. average values).
We argue that continuity and discontinuity in special relativity play an important role in the probabilistic wave properties of quantum mechanics. Special relativity is involved with transformations from a rest frame to a frame moving with a constant -v, for example . The discontinuity appears in the form of a change from v=0 to v, while the continuity arises from Lorentz invariants which indicate one has the same physical state whether it is viewed from a frame with one velocity or another. This invariance applied to E, p (momentum) as a modulus forces a minus sign between the two, in order to allow for linkage with E=mo (c=1) and p=0, i.e. -EE + pp = -momo (c=1). Another invariant, however, is A= -Et’ + px’, and in this case the minus sign and the discontinuity lead to a new x’,t’ (in comparison with the rest frame for a particle with rest mass mo), but also to t’ -> t’+ b/E and x->x’ +b/p leaving A unchanged (b=constant). The discontinuity in v, which leads to a jump in x’,t’ (as seen in the moving frame) also manifests itself in a time-length interval which ultimately leads to exp(-iEt+ px), we argue. In other words, classically one considers constant velocity as smooth continuous motion, but the introduction of two frames, v=0 and -v, leads to a discontinuity which leads to “chunkiness” in the form of a frequency and wavelength. If these cannot be perceived classically (as is often the case), then one considers only smoothness. Relative differences in time or length or in frequency or wavelength, however, can appear in nature for length scales of the order of the frequency/length scale. In the case of n-slit interference-diffraction, a relative difference leads to a “jump” (so to speak) which manifests the wavelength. In particular, a photon moves along the y direction with a phase px (considering a steady state picture). It reaches a two-slit system along the x-axis and there is probability for it to interact with both slits. If one considers two parallel rays from each slit moving towards a point on a screen very far away (as is done traditionally), there is again smooth motion with a rotated p vector (same magnitude), but for a point on the screen to the left of the center, there is a jump in the length associated with the right slit. In other words, there is a jump similar to the v=0, -v jump. This leads to a relative difference in length associated with both rays (linked to an OR probability situation using exp(ip dot r). This relative difference in length displays the phase difference initiated by special relativity. In the case of a bound state, different impulse hits by a potential create different momenta which have different phases exp(ipx) because of special relativity and one has interference. The same phase A= -Et + p dot r, which is linked with a chunkiness due to a discontinuity, is also associated with continuity because A is a Lorentz invariant. Thus A=constant may be used to solve for refraction/reflection problems with or without moving mirrors/interfaces with different indices of refraction. Continuity appears further in a 1-dimensional reflection/refraction problem when applied to the exp(ipx), i.e. Aexp(ipx) + Bexp(-ipx) = C exp(ip2x) where E=pc = p2 c2, p2=n2p and c2=c/n2 with n1=1, together with a continuity equation for the first derivative. Thus, both continuity and discontinuity (or jumps), which follow from special relativity, seem to play an important role in quantum mechanics.