Classical statistical mechanics is based on the notion of a particle (i.e. its x and p phase space) as is Newtonian mechanics which is based on x(t). Quantum mechanics, on the other hand, contains particle-wave duality and is linked to Newtonian mechanics (in that one considers bound state, motion etc), but is also considered to be statistical. In this note we consider the statistical features of quantum mechanics by examining a reflection/refraction of light problem with incident light moving along the positive x direction. If considered from the point of view of a particle (i.e. photon) (in accordance with Newtonian mechanics and classical statistical mechanics) there is no equilibrium. One may imagine one photon strike the medium at a time. This photon retains its energy, but has a certain probability “a” to reflect at x=0 and 1-a to refract. Thus energy and photon number are conserved, but “a” is unknown and the energy and particle number to the left and right of x are not the same, even if one considers a steady state stream. In other words, there is a discontinuity at x=0 because of the difference in energy and photon numbers, yet a steady state picture seems to imply a kind of equilibrium which is not present in the particle picture. In other words, there should be invariance along the x axis with continuity at x=0, but this cannot be achieved by only considering particles (i.e. photons). Nevertheless, photons are physical and whatever solution is found must be consistent with photon energy and number conservation. This seems to suggest a new physical property needed to establish the equilibrium which somehow coexists with the particle picture. We suggest this property is wavelength with a photon being described by exp(ipx) (the square root of flux) and that it is this scenario which allows one to see the reflection/refraction problem as an equilibrium one. In other words, one does not need a particle point of view to establish equilibrium, yet particle number/energy may still be conserved in this picture.
Classical statistical mechanics is based on the notion of a particle (i.e. its x and p phase space) as is Newtonian mechanics which is based on x(t). Quantum mechanics, on the other hand, contains particle-wave duality and is linked to Newtonian mechanics (in that one considers bound state, motion etc), but is also considered to be statistical. In this note we consider the statistical features of quantum mechanics by examining a reflection/refraction of light problem with incident light moving along the positive x direction. If considered from the point of view of a particle (i.e. photon) (in accordance with Newtonian mechanics and classical statistical mechanics) there is no equilibrium. One may imagine one photon strike the medium at a time. This photon retains its energy, but has a certain probability “a” to reflect at x=0 and 1-a to refract. Thus energy and photon number are conserved, but “a” is unknown and the energy and particle number to the left and right of x are not the same, even if one considers a steady state stream. In other words, there is a discontinuity at x=0 because of the difference in energy and photon numbers, yet a steady state picture seems to imply a kind of equilibrium which is not present in the particle picture. In other words, there should be invariance along the x axis with continuity at x=0, but this cannot be achieved by only considering particles (i.e. photons). Nevertheless, photons are physical and whatever solution is found must be consistent with photon energy and number conservation. This seems to suggest a new physical property needed to establish the equilibrium which somehow coexists with the particle picture. We suggest this property is wavelength with a photon being described by exp(ipx) (the square root of flux) and that it is this scenario which allows one to see the reflection/refraction problem as an equilibrium one. In other words, one does not need a particle point of view to establish equilibrium, yet particle number/energy may still be conserved in this picture.
Classical statistical mechanics is often presented in terms of the maximization of entropy (ln(# arrangements) or ln( total probability for a set of arrangements)) subject to a constraint. In a previous note (1), we argued that such a process seems to be associated with a distributive variable which is a scalar i.e.is additive such that a product of probabilities Product over i n(ei) of different values of this variable Sumover i ei = E is equivalent to a single probability of E i.e. n(E). We applied this idea to both a Maxwell-Boltzmann gas and a 3D random walk. This distributive variable changes in the stochastic process of the problem. For example for an MB gas, there are elastic collisions so energy is the distributive variable. For an MB gas in a potential V(x), there is potential and kinetic energy of a single particle, but this is not regarded as a stochastic process in the classical view. In (2) we argued that a distributed variable is not the only type of equilibrium which may exist, even in the classical world. We considered the example of a beam of light which reflects and refracts at x=0 (medium along the y axis). Energy is conserved even at the photon level, but there is an unknown probability namely that for reflection “a” because the refraction probability is then 1-a. Thus energy conservation is not enough. We argued that in order to solve for this problem in an equilibrium or steady state manner, one needed to deal with square roots of fluxes/intensities in order to obtain two equations to solve for B,C i.e. square root fluxes at x=0 for the reflected and refracted rays in terms of A, the square root flux for the incident beam. We argued these fluxes were associated with eigenfunctions of the translation generator i.e. exp(ipx). Thus a dynamic equilibrium is introduced (i.e. quantum mechanics) which does not use a distributed variable i.e. does not use maximization of entropy. Rather it is associated with eigenfunctions of the generator of the dynamical motion involved. It is this dynamical variable (e.g. p) which changes in this stochastic process even if in direction only. A question arises: When does one use one type of equilibrium approach versus the other? We attempt to answer this 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?
Netwonian mechanics and quantum mechanics seem to be very different theories, even though Newtonian concepts like momentum, energy and their conservatioon carry into quantum mechanics. The difference is not even due so much to the case that Newtonian mechanics is deterministic and quantum mechanics, statistical, we argue. We suggest that quantum mechanics utilizes a turly dynamic probability which treats forwards and backwards motion differently thorugh exp(ipx) and exp(-ipx), while Newtonian mechanics treats them in essentially the same manner, with the exception of the direction. We thus suggest that Newtonian mechanics is really based on the notion of stationary objects. In particular, x(t) means that a particle is at x in dx at t, as if it were stationary with the exception that it is translating with a fixed velocity. This stationary viewpoint might describe why the P(x)=1/L probability for stationary objects is sometimes also applied to objects moving with constant speed. The x(t) approach really means at t, the particle is frozen at x for dt before appearing frozen at x+dx at t+dt with a new velocity. There is of course the notion of acceleration, but for a particle to be at x at t, one must measure it and this means that it is as if it is stationary for an instant. The velocity is then considered to be constant at x (i.e. a translation) with a new translation value at x+dx. As a result, the notion of quantum mechanics does not really follow directly from this Newtonian stationary viewpoint. One has to think in terms of a very different idea, namely that exp(ipx and exp(-ipx) are not merely translations in different directions. A truly dynamic theory is needed. As we have argued in previous notes, P(x)=1/L represents a stationary object and so if one had an AND condition for p and -p (i.e not speed), then exp(ipx)/sqrt(L) exp(-ipx)/sqrt(L) = 1/L. Freezing out speed using quantum dynamic probability exp(ipx) (exp(-ipx)) leads to a stationary picture, but the actual physics of exp(ipx) is very different from anything in Newtonian mechanics. This leads to the question: What happens in the high energy (i.e. classical limit) if Newtonian mechanics and quantum mechanics are so fundamentally different? Quantum mechanics differs from classical physics in that there are fixed uncertainties of dt = hbar/E and dx=hbar/p for a particle with E,p, while Newtonian mechanics uses infinite resolution. One may note that exp(ipx) and exp(-ipx) are very different, but that is only if one can resolve x regions on the order of hbar/p. If one has worse resolution, one would not notice that exp(ipx) and exp(-ipx) differ. They are both possibly present in a dx which is the resolution limit. Thus, it is actually not the infinite resolution of Newtonian mechanics that is manifest, but rather the approximate finite resolution which means that on classical scales, a tiny dx is actually larger than hbar/prms where prms is the root means square momentum of a quantum bound solution. The quantum bound state solution is based on finding energy levels and a wavefunction W(x) = Sum over p a(p)exp(ipx). It includes exp(ipx) and exp(-ipx), but physically in the classical limit, the particle moves to the right for some time and then later to the left. Quantum mechanics cannot change this measured situation. As noted, exp(ipx)*exp(-ipx) =1 and this is linked with a stationary object even though exp(ipx) and exp(-ipx) both represent dynamic probabilities. The dynamic probabilities allow one to correctly solve problems at scales of hbar/p. W*(x)W(x) is an attempt to create a stationary distribution which is consistent with the kind of measurements made in the classical world. It is W(x), however, that is actually used in solving problems. As the energy level increases, the peaks of W*(x)W(x) become increasingly narrow and so even this stationary-quantum approach is replaced by the simpler, approximate exp(i prms x) * exp(-i prms x), leading to a stationary free particle distribution. This is why we suggest that Newtonian mechanics is really a stationary free particle solution with the particle then translating from one x (within dx) to the next in dt. In other words, x(t) means that the particle is at rest in t (for an instant as it must be measured). The v(x) = dx(t)/dt is called the constant speed in dx (as it must be measured) even though there is acceleration occurring. It seems that Newtonian mechanics is a static kind of viewpoint which is very different from the dynamic exp(ipx) and exp(-ipx).
Quantum mechanics is often compared with classical mechanics because both may deal with a single particle subject to a potential. Classical mechanics is said to be deterministic describing x(t) and its derivatives i.e showing how a particle moves and accelerates while bound state quantum mechanics produces a real wavefunction W(x) such that W(x)W(x) yields a spatial density profile. The comparison is carried further with the correspondence principle which states that at high energy levels quantum mechanics should mimic classical mechanics. Classical mechanics does not simply describe x(t) and its derivatives i.e. deterministic motion. It also includes two body scattering, force and torque balances for a stationary object etc. In the case of the classical statistical mechanics of an equilibrium ideal gas, the Maxwell-Boltzmann (MB) distribution in the absence of a potential is C1 exp(- pp/(2mT) ). If a potential V(x) is added, one may find P(x) two different ways using Newtonian ideas. The first is to consider the gas as a collective entity like a tiny macroscopic classical object at x. Then one may argue that equilibrium means the system is stationary and apply Newtonian statics i.e. a balance of pressures and -dV/dx to obtain exp(-V(x)/T) as is often done in textbooks. Alternatively, one may consider a single particle and apply Newtonian mechanics differently, namely use a conservation of energy equation: p1p1/2m + V(x1) = p2p2/2m + V(x2). Again one obtains exp(-V(x)/T). The spatial part of probability may be based on Newtonian mechanics indicating that it is not stochastic. The stochastic function is C1 exp(-pp/2mT). Interestingly, however, the two Newtonian approaches are different with respect to a single particle. A collective nonmoving set of particles acts as a stationary object, yet within it each particle seems to be “bouncing” off of an imaginary wall creating pressure. Thus one has an impulse free particle picture yet obtains the correct result exp(-V(x)/T). Alternatively one may think of a single particle accelerating which it does if it is not colliding with other particles. This is a very different picture of the single particle than of one bouncing back and forth hitting an imaginary wall. In fact, there is usually a wall holding the gas together and pressure and bouncing particles are the real picture there, but within the gas one would normally think of a single particle accelerating. Nevertheless there are two scenarios which yield the same result. The idea of pressure balance is more in keeping with an overall statistical picture of the gas. In other words, one may actually ignore the idea of following a single particle as it accelerates. (That is not to say that single particle acceleration does not exist - it does.) Quantum mechanics we argue is a statistical theory. In the case of a single bound particle one has a momentum distribution W(x)=wavefunction = Sum over p a(p)exp(ipx). One may apply conservation of average energy at each x to obtain the time-independent Schrodinger equation. In such a picture, like with the statistical mechanical gas, we argue that one may imagine impulse hits with the potential instead of considering acceleration. The quantity vrms(x) from KEclassical= .5m vrms(x)vrms(x) = -1/2m d/dx dW/dx / W does display acceleration, but this is a mathematical average. As energy levels become higher the correspondence principle is said to apply, but a classical particle exhibits acceleration. The quantum bound state does not, but like in the case of the statistical mechanical gas it may be that two scenarios lead to the same result. In other words, the quantum bound W(x) still has crests and troughs, but with low resolution one may only see an envelope roughly touching W(x)W(x) which matches a classical C2/v(x) where v(x) is classical velocity. Given that quantum mechanics is a statistical theory it may not be necessary to have it explicitly display acceleration.
In (1), we argued that a uniform distribution associated with the placement of static particles in a one-dimensional length L is P(x)=1/L. We then suggested that because a static particle has speed=0, one cannot rule out the existence of a probability F(p,E,x,t) which is Lorentz invariant. For every p, there is a -p and so F(p,E,x,t) complex conjugate * F(p,E,x,t) = 1/L yielding F(p,E,x,t) = exp(-iEt+ipx)/sqrt(L) (see (1) for details). F(p,E,x,t) is a dynamic probability like x(t) or v(x,t) are dynamic variables in Newtonian deterministic theory. We then argued that the F(p,E,x,t) is not simply a math result, but actually solves the 1-D reflection-refraction problem, while Newtonian mechanics cannot. Here we ask the question: Why can’t Newtonian mechanics solve the 1-D reflection-refraction problem? Certainly there is probability in the problem, given by 1= probability(reflect) + probability(refract). It seems that Newtonian mechanics requires that one always know x(t), v(x,t) etc. Imagine that there is a specific deterministic mechanism for reflection and a different deterministic one for refraction and that these turn on with probability(reflect) and probability(refract). One may then describe reflection and refraction separately by deterministic Newtonian mechanics, but one cannot calculate probability(reflect) and probability(refract) because these are built into the physical interfaction interface n1-n2 (where ni is the index of refraction). Experimentally, the probabilities to reflect and refract relative to the incident beams are functions of only n1 and n2. N1 and n2 are known, but that still does not provide one with a method of solving for p(reflection) and p(refraction).Newtonian mechanics is not of use. The question then becomes: What if one introduces a dynamic probability exp(ipx), dropping exp(-iEt) because E is the same for the incident, reflected and refracted beam? How could this possibly help matters? The probability to reflect and refract is based on n1 and n2 alone. We consider the notion that exp(ipx) is a dynamic probability and it seems that it may also be thought of in terms of a steady stream. A steady stream of particles means that one does not know the exact x(t) of any particle, one only has an average sense of the probability flow. This suggests that one can use an extra conservation law that is not available in Newtonian mechanics, namely conservation of dynamic instantaneous probability exp(ipx). We stress that 1-D reflection is not a physical fluid like steady stream problem. We use the notion of a steady stream as an analogy to describe the conservation of exp(ipx), the dynamic probability. Given that exp(ipx) is an instantaneous probability, it must be accounted for at every x point and the viewpoint of a steady stream allows one to bypass single reflection or refraction events because one is viewing the process on average, but at the level of the instantaneous probability which must be defined at each x. This is almost like having probability represent fluid flow, but it is the dynamic instantaneous probability which behaves like the fluid, not a classical flux. Such a new conservation law (and conservation of the dynamical average of momentum) yields two equations which allow one to solve the problem. In a nutshell, Newtonian mechanics does not allow one to combine events that occur at different times in the same equation as if they occurred at the same time. The OR (addition) probability condition allows for this, but 1= probability(reflect) + probability(refract) does not contain the dynamics of the event. One requires a probability which acts like a steady stream probability and we argue that this is the dynamic probability exp(ipx). This forces one to deal with the probability being conserved in x, but also momentum being conserved under this scheme (i.e. using this probability) as exp(ipx) is the instantaneous probability describing a particle with momentum p.
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.