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).
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 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.
Newtonian mechanics deals with smoothness in time and space through x(t) where the “details” occur within dx and dt which tend to zero. Thus there are no discontinuities in x and t. We have argued that the quantum free particle wavefunction pieces exp(-iEt) and exp(ipx) deal with discontinuities and that smoothness is due to a “modulus” of two dimensional probability e.g. exp(ipx)exp(-ipx) which yields 1 or determinism. Both exp(-iEt) and exp(ipx) contain finite measures in time and space, namely hbar/E and hbar/p and so there may be discontinuities. For example, exp(ipx) contains probability cos(px) and sin(px) within a wavelength so x=vt is only an average type of motion. Thus using exp(ipx) in a problem may result in time discontinuities because the particle is not moving as x=vt within a wavelength. In Part I we examined one-dimensional reflection/refraction of light from an interface of two media with n1=1 and n2>n1. We concluded that the quantum solution of this problem involves two equations, one linked to probabilities and the other to impulse. Multiplying the two equations together (with a special grouping, namely a spatial grouping) yields a single equation which may either be interpreted as a pressure balance equation. Thus one starts with two equations which are both grouped in time, but with the second demonstrating vector directions and obtain a single equation which is only grouped in space (which is linked with spatial grouping). By rearranging this equation, one may obtain a time grouping i.e. energy conservation equation, but the two groupings cannot exist together. This is the problem with the classical pressure picture linked to a steady state scenario and not a single photon process which displays a discontinuity in time i.e. a single photon may either reflect or refract. In this note we try to consider why classical energy conservation/ pressure balance only leads to one equation, while the quantum mechanical approach leads to the two required equations at the price of introducing a square root flux type of probability. We argue that there seem to be two factors. First there is the notion of scalars and vectors and a space versus time picture. The single conservation of energy equation AA/c1 = BB/c1 + CC/c2 where AA,BB,CCC represent the incident, reflected and refracted beams is expressed in terms of time and so vector information (i.e. -p for the reflected beam is not conveyed). If one rewrites this as AA/c1 -BB/c1 = CC/c2 then one obtains a pressure balance type of equation, but time is removed and one is grouping by space. The second and more important feature we argue is time discontinuity. If one considers a single photon, it may reflect or refract, but not both. Thus there is a time period in which to observe both reflection and refraction and more than one incident photon is needed. Newtonian mechanics in the form of steady state treats time as smooth and so everything must happen instantaneously. This is the idea behind pressure. One considers an impulse 2p multiplied by flux i.e. velocity. The flux is used to find the number of particles which strike a surface per second as if this were a continuous smooth process in time. For the case of reflection/refraction it is not. It is discontinuous because either a reflected ro a refracted photon is created each time. Thus the notion of pressure averages this over and loses some of the information in the problem. This, we argue, is why a pressure balance/ energy conservation equation is not sufficient to solve the reflection/refraction problem even though pressure is a physical concept and may be measured (on average) just as x=vt may be measured on average.
We have argued in previous notes that quantum formalism emerges from the one dimensional photon reflection-refraction scenario which is readily observable in the classical world. What about the situation for a particle with rest mass? In previous notes, we have considered the case of a constant E,p and V1 which changes at xo to E,p2,V2 (all again constants with E remaining unchanged). This problem is then associated with an incident and reflected particle for x<xo and a particle with p2 for x>-xo. In a classical world, however, how does one obtain reflection and refraction together? If there is a strong wall, a particle may reflect with -p and the same kinetic energyE, but it does not pass through the wall. If there is a ramp at xo leading to a horizontal floor a little higher than that for x<xo, then one has a conserved total E, but a new p2 and V2. The ramp, however, is not a sudden change at xo, but occurs over a range of x. As a result, it seems the quantum scenario, rather than leading to the classical case in a high energy limit (as in bound states), is associated with fluctuations and non-sharp boundaries. It is fine to have a sharp wall at x=xo, but in quantum mechanics, one has atoms with electrons moving in space according to probability distributions. Furthermore, there is movement due to temperature. One does not have a sharp edge or ramp scenario alone, but fluctuations which could lead to both. This suggests stochasticity at a point of discontinuity, but also stochasticity in the regions V1 and V2. In other words, V1, V2 and the sharp discontinuity points are all approximations (averages). If one accepts the fluctuations at xo (ramp and sharp wall), then classically one might impose a pressure balance equation at an idealized xo, i.e.: AAp -BBp=p2CC where AA represents a positive number etc. AA represents the incident particle, BB the reflected and CC the particle which moves up the ramp. Nonrelativistically and in the photon case, the pressure balance equation is also equivalent to a conservation of particle number or energy equation. Thus one has two unknowns (if A is known), i.e. B and C, and one equation. Here, however, we wish to stress a different, but related issue. The pressure balance equation allows for multiple solutions because it only exists at xo. For example, BB=0 is a solution as is BB=.5, CC=.5. If p2 is almost equivalent to p, one might expect that BB=0. In other words, there is a specific solution which goes beyond pressure balance, i.e. here seems to be a p (momentum) biased solution. This forces one to go beyond classical mechanical ideas, even if one introduces a fluctuating wall-ramp scenario at xo. The fluctuating scenario suggests fluctuations in V1 and V2 as well, suggesting one may have some kind of dynamical equilibrium in these regions. Would this be enough to create a momentum bias, linked say to a continuity of a function of p,x and its first derivative at xo? This extra momentum bias seems to be the extra feature which emerges in a quantum treatment of the problem, which only occurs because one introduces fluctuations at the xo point. We suggest that a dynamical equilibrium approach solves the problem if one maximizes Shannon’s entropy subject to a the constraint of a constant average classical action written as: -P(t) iEt + ipx P(x) with a Lagrange multiplier of 1. “i” is introduced to a have a periodic/dynamic probability, i.e. one that does not grow. Thus we argue that quantum mechanics for a particle with rest mass emerges because a classical pressure balance equation for a fluctuating wall-ramp picture at a V1-V2 junction allows for multiple solutions. Ifr V1 is almost equal to V2, one has a biased or selective solution which should force BB to be 0. This may be done through continuity, but there must be a physical basis for the function used and we suggest that this is maximization of Shannon’s entropy subject to an average classical action introduced to make the probability periodic due to the dynamical nature of the problem. Thus there is not just stochasticity at the discontinuity point xo, but throughout x.
Newtonian force is based on a change in momentum in time, but momentum is a vector and so may change in magnitude and direction. In many cases, the change is of one form or the other. For example, for a bound particle the change is in kinetic energy i.e. in magnitude, while for a particle moving at constant speed in a circle or pressure in a Maxwell-Boltzmann gas, the change is in direction. For an MB gas, the probability for p and -p are the same so the average momentum at any point x is 0 and thus chunks of the gas do not move, even though a single particle in the gas moves. The pressure, on the other hand, is the average of 2pv (2p for impulse, v for flux) and so pressure for p and -p do not cancel. A quantum free particle is based on exp(ipx) which does not contain time and is a dynamical probability, This is both an eigenfunction of the momentum operator -id/dx and the kinetic energy operator -1/2m d/dx d/dx, yet as pointed out in Part II, there are two pieces (cos(px),sin(px)) with a spatial density and hence a kind of pressure/force. These, however, combine to give the sense of constant motion in one direction. We argue this is due to a dynamical probability, unlike a static balance of pressures/forces found in an MB gas or Newtonian statics. There is a density with pressure distribution cos(px), but this is continuously changing because the particle moves in one direction with constant p. Thus the two pressure distributions “balance” so to speak to create a constant p. Interference of exp(ipx) and say exp(-ipx) may “bring out” a “static” spatial density i.e. 2cos(px), but behind the scene the dynamical probabilities still exist. It is these which “hold” the cos(px) together which appears strange from a static point of view. For example, a quantum bound state has a static W*(x)W(x) spatial density at the same time that the single particle has an average classical kinetic energy KE(x)=En-V(x). Interference also gives rise to the notion of an impulse-pressure in quantum as one ignores the isin(px) and i sin(-px) terms. Creating a pressure scenario with p and -p leads to two considerations. First, there is an impulse-pressure (defined without flux) at each x point and secondly, this impulse-pressure changes with x. The time-independent Schrodinger equation is formulated in terms of energy (i.e. a scalar): KE(x)+V(x)=En where KE(x)= {-1/2m d/dx dW/dx} \ W and W=Sum over p a(p)exp(ipx). Thus, as in Newtonian mechanics, one thinks of -dV/dx solely accelerating a particle. The quantum particle however, is described by an ensemble W(x) which has pressure. This pressure is due to a combination of the form exp(ipx) and V(x), so V(x) does not simply change root mean square speed. exp(ipx), by itself, creates pressure when interference occurs as in a two slit scenario. We try to examine these ideas in this note.
We suggest that quantum mechanics is concerned with preserving two conservations, one related to a kind of probability and the other to a mechanical balance.These two conservations exist at the level of a single particle and are interlinked. Specifically, the derivative d/dx, which measures the effect of a change in a kind of probability, also measures the associated resistance or impulse for a free particle. For a free particle, the “kind of probability” is exp(ipx) and d/dx exp(ipx) = ip exp(ipx) showing that there is an impulse related to p linked with the probability. In other words, a free particle moves through space in a probabilistic manner related to the resistance associated with the impulse p. These two conservations, a kind of probability and a kind of resistance, lead to two independent equations when one considers continuity at points which change momentum. We consider again the problem of a photon moving in the x direction which reflects or refracts at an interface along the y axis between n1-n2 (indices of refraction). Classically, probability is conserved i.e. 1= P(reflect) + P(refract) ((1)). The steady state problem, however, has the appearance of an equilibrium and should have a second conservation, namely one in space, i.e. a balance of pressure. This balance, linear in momentum, however, is the identical equation as ((1)). Thus, there are still two unknowns P(reflect), P(refract) and one equation. In the quantum case, double conservation may classically be manifested by the same equation. This seems to suggest, we argue, that ((1)) is a consequence of an equation based on p (which is linked to momentum). We try to find this association in two ways. First, we argue that the interface may exist at xo and that there are various momenta in the problem, We argue these represent pertinent information which does not appear in ((1)). We suggest that ((1)) seems to be a kind of magnitude equation with p and xo information being part of the direction of vectors i.e. p and xo may exist in a phase exp(i f(p,x)). Together with the conservation of pressure condition, we argue that exp(i f(p,x)) = exp(ipx). This argument, however, implicity involves consider the phase function exp(if(p,x)) as a kind of probability. Alternatively, we consider finding this probability follows from maximizing Shannon’s entropy subject to the periodic constraint i p x P(x), but we investigate why Shannon’s entropy form should be used, i.e. what is its physical significance?
In a previous note (1), it was shown that for a ground state oscillator wavefunction, one could analyse the problem in momentum space as a free gas of momentum particles and obtain exp(-p*p/2mT) as fp*fp, fp being the momentum wavefunction. A free energy equal to zero was obtained, with kinetic energy being the energy term in E-TS. This suggests quantum particles scatter (with conservation of momentum) like classical point particles in momentum space, but differ from classical point particles in position space. In particular, one takes a Fourier transform of fp as it seems that each momentum wave carries an exp(ipx) function which is a sort of “baggage” term, possibly due to zitterbewegung. This seems to explain why a free gas in momentum space for the ground state oscillator yields a Gaussian density with W(x)=exp(-ax*x) (W(x)=wavefunction). An oscillator potential is then needed to keep this density with its momentum space pressure in balance. Thus, quantum mechanics and classical statistical mechanics become very similar for this scenario. This begs the question: What about other potentials or oscillator states higher than the ground state? In such a case, it still seems that one may consider “particles colliding in momentum space” with each carrying an exp(ipx) “baggage” term, but one does not have a “classical statistical mechanical” equilibrium scenario any longer. Rather, it seems that one has a net force created by -dV/dx (V(x)=potential) and a difference in pressure d/dx [Average(p*p)d(x)] where d(x) is density. There also seems to be an average velocity (1/W)dW/dx. In momentum space such a velocity would not exist for f(p)=f(-p), but because of the baggage of exp(ipx) it is present i.e. p exp(ipx) - p exp(-ipx). Newton’s law, with force acting on this velocity appears in the equation d/dx [Average(p*p)d(x)]=F(x) d(x). This same equation is equivalent to the gradient of the time-independent Schrodinger equation. For quantum mechanics, one has one particle rather than many as in classical statistical mechanics. Thus, it seems that this “Newton’s law” balance is based on a long term time average. It seems, however, that there should be a period T during which the average can establish itself. After this, one would have periodic time motion where the system keeps scattering back into itself in this characteristic time. This period can be deduced, it seems, from the time-independent Schrodinger equation and is found to be E, the energy.