In classical statistical mechanics, e.g. a Maxwell-Boltzmann (MB) gas, one tries to find the probability distribution of a particular quantity such as particle energy pp/2m. Thus there is a distribution in a quantity, making it the stochastic variable. This quantity, however, is subject to a conservation law i.e. elastic two particle scattering. In a previous note (1), we considered the case of reflection/refraction and called momentum the stochastic variable and noted that it too was linked with conservation. In fact, we argued that both the MB gas and reflection/refraction followed the same stochastic law: P(variable1) P(variable 2) = P(variable 1 + variable 2) whether variable 1 is a scalar or vector. In (2), however, we noted that exp(ipx), a free particle wavefunction, treats p as a parameter with x being the “probability” variable. This seems a little unusual because p, which is normally a variable in an MB gas and even in a quantum bound state i.e. a*(p)a(p) where W(x)=wavefunction = Sum over p a(p) exp(ipx) takes on the role of a parameter i.e. p=hbar/wavelength. Why this duality? Why is p a parameter for “x” of the probability distribution and yet at the same time a variable associated with conservation of momentum? In this note, we argue that unlike the distribution of a defining property of a particle such as momentum or energy, one may have a distribution associated with probability itself. This we argue is the case for a reflection/refraction problem which applies to both light and a particle. In the case of light, there is an interface of two media with different indices of refraction and in the case of a particle an interface with two constant potentials. In both cases, total energy of the photon/particle is constant, but there are two choices or probabilities, reflection or refraction, but they are linked with different times. A single photon or particle can only undergo one at a time. One knows exactly the two values of momentum, but the probability of each is unknown. Thus unlike an MB gas where energy/momentum is being redistributed because the two particles of the outcome exist at the same time, here a probability of 1 must somehow be distributed to the two events.. Given that the incident momentum is p, the reflected -p and the refracted p2, only considering a specific time event i.e. reflection or refraction, does not conserve momentum. In the MB case, a single time event i.e. scattering conserves both momentum and energy. Thus it seems that one does not deal with events in time. There is a distribution of probability in such a way as to ensure conservation of momentum, but does this probability already exist or get created at the interface point? We argued in (3) that if the scenario were deterministic one would have to follow it in time and treat each event separately. Rather we suggest that a distribution of probabilities allows for conservation of momentum, but this means that all three momenta exist at the interface x point as one is not considering an event at a precise point in time. This then allows for each momentum to have its own x distribution based on momentum p so that one may impose conservation of momentum and not have a deterministic time associated with the interface point.. This last idea of conservation of momentum appearing is not discussed in (3) and allows for p to behave like a parameter for x while at the same time being a variable which is conserved. In short, in an MB gas one distributes energy between two colliding particles in order to conserve total energy while for a quantum particle or photon which reflects/refracts one distributes probability to the two scenarios (reflection or refraction) such that momentum is conserved. Thus the nature of what is “distributed” is very different.
In classical statistical mechanics, e.g. a Maxwell-Boltzmann (MB) gas, one tries to find the probability distribution of a particular quantity such as particle energy pp/2m. Thus there is a distribution in a quantity, making it the stochastic variable. This quantity, however, is subject to a conservation law i.e. elastic two particle scattering. In a previous note (1), we considered the case of reflection/refraction and called momentum the stochastic variable and noted that it too was linked with conservation. In fact, we argued that both the MB gas and reflection/refraction followed the same stochastic law: P(variable1) P(variable 2) = P(variable 1 + variable 2) whether variable 1 is a scalar or vector. In (2), however, we noted that exp(ipx), a free particle wavefunction, treats p as a parameter with x being the “probability” variable. This seems a little unusual because p, which is normally a variable in an MB gas and even in a quantum bound state i.e. a*(p)a(p) where W(x)=wavefunction = Sum over p a(p) exp(ipx) takes on the role of a parameter i.e. p=hbar/wavelength. Why this duality? Why is p a parameter for “x” of the probability distribution and yet at the same time a variable associated with conservation of momentum? In this note, we argue that unlike the distribution of a defining property of a particle such as momentum or energy, one may have a distribution associated with probability itself. This we argue is the case for a reflection/refraction problem which applies to both light and a particle. In the case of light, there is an interface of two media with different indices of refraction and in the case of a particle an interface with two constant potentials. In both cases, total energy of the photon/particle is constant, but there are two choices or probabilities, reflection or refraction, but they are linked with different times. A single photon or particle can only undergo one at a time. One knows exactly the two values of momentum, but the probability of each is unknown. Thus unlike an MB gas where energy/momentum is being redistributed because the two particles of the outcome exist at the same time, here a probability of 1 must somehow be distributed to the two events.. Given that the incident momentum is p, the reflected -p and the refracted p2, only considering a specific time event i.e. reflection or refraction, does not conserve momentum. In the MB case, a single time event i.e. scattering conserves both momentum and energy. Thus it seems that one does not deal with events in time. There is a distribution of probability in such a way as to ensure conservation of momentum, but does this probability already exist or get created at the interface point? We argued in (3) that if the scenario were deterministic one would have to follow it in time and treat each event separately. Rather we suggest that a distribution of probabilities allows for conservation of momentum, but this means that all three momenta exist at the interface x point as one is not considering an event at a precise point in time. This then allows for each momentum to have its own x distribution based on momentum p so that one may impose conservation of momentum and not have a deterministic time associated with the interface point.. This last idea of conservation of momentum appearing is not discussed in (3) and allows for p to behave like a parameter for x while at the same time being a variable which is conserved. In short, in an MB gas one distributes energy between two colliding particles in order to conserve total energy while for a quantum particle or photon which reflects/refracts one distributes probability to the two scenarios (reflection or refraction) such that momentum is conserved. Thus the nature of what is “distributed” is very different.
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.
In Newtonian mechanics a particle with constant momentum moves deterministically according to x=p/m t. Even a quantum object such as a photon or electron should satisfy this equation on average to give meaning to the notion of the speed of light in a vacuum, for example. Thus it seems surprising that a quantum free particle is also associated with exp(ipx) which only contains momentum and is periodic and complex. Furthermore the physical consequences of exp(ipx) are real as shown in a two-slit interference experiment etc. In this note, we attempt to discover why exp(ipx) exists in the first place by examining the example of the reflection/refraction of light which occurs in the macroscopic classical world. Basically our argument is based on analyzing two important pieces of information in the system, namely the scalar energy (or particle count) and the vector momentum. A main point we make is that energy or particle number considerations are linked to determinism i.e. a specific time of interaction. One may note that the interaction x is fixed by the two medium barrier. This means a single incident photon exists in one time range and the reflected OR refracted photon in another. It is possible to compare the incident photon to either the reflected or refracted or to a sum of probabilities. In particular, a single photon has an energy E and a count of 1. If it reflects, it has an energy E and count of 1 so there is a balance, likewise for refraction. Thinking in terms of probabilities “a” and 1-a to reflect/refract: 1= a + 1-a. Can one perform such a matching for momentum, which is a physical piece of information? We argue one cannot. For example, a single incident photon has a momentum p which is linked to the impulse it may deliver. If it refracts, “all of a sudden” it is linked with momentum of p2 = n2 p where n2 is the index of refraction in the second medium and n1=1. This seems problematic because a physical impulse of 2p is all of a sudden an impulse of 2p2. Thus there exists a big problem which seems to require the notion of probability and all three momentum values p, -p and p2 considered together, unlike the situation of energy/particle number for which a comparison of the incident photon and one possible result yields a balance. Combining all three momenta, however, is also problematic in a deterministic picture because the incident photon exists in one time range and the reflected or refracted in another. How can one possibly compare them together? The solution seems to be to argue that the incident photon is only deterministic on average. In particular, if it has an x probability distribution over a certain length (called a wavelength), one does not know the precise time at which it hits “x” the medium interface. In such a situation, there is time uncertainty which allows all three momenta to be compared at the same x. This, however, leads to a major consequence, namely that so-called deterministic motion associated with constant p is deterministic on average, but contains repeated pockets of x probability distributions such that time is not deterministic within. We try to examine these ideas in this note.
There is a focus in classical statistical mechanics on maximizing an entropy function subject to constraints to obtain probability values. For a coin or die probabilities are given, yet one may still calculate a Shannon’s entropy. In the case of a Maxwell-Boltzmann (MB) gas, one may obtain probabilities using energy conservation and products of probabilities and then show this is equivalent to maximizing an entropy function subject to an energy constraint. We argue that the energy conservation approach yields an energy conservation equation and a product of probabilities representing a certain total energy equated to a product of different probabilities representing the same total energy. The two equations are linked to solve for probability in terms of energy and this is formally the same as a maximization problem subject to conservation of energy. A point we make is that this approach is based on a kind of static conservation, even in the presence of a potential V(x). In quantum mechanics, we argue one one has fluxes even though there is also conservation of energy. For light moving in the positive x direction which reflects and refracts (with each photon retaining its energy) there is both energy conservation and flux. The energy conservation equation in terms of flux/intensity terms is given in (1) as: AA/c1 = BB/c1 + CC/c2 where A,B,C refer to the incident, reflected and refracted rays. This single equation cannot solve for probabilities B,C in terms of A and there is not a separate product of probabilities equation as in the MB case. Rather quantum mechanics seems to be solved using square root fluxes as dynamical probabilities. In particular, the energy conservation equation is equivalent to two square root flux equations A+B=C and p1A -p1B = p2C where E=pc. Thus probabilities are determined through dynamics not static conservation or entropy maximization. In an earlier note we concluded that exp(ipx), a quantum free particle wavefunction, is also like a square root flux, and also acts as a dynamic probability. These ideas may be extended to a quantum bound state which we discuss in the note. Thus the approach to finding probabilities in the classical statistical mechanics and quantum realms seem to be completely different. One uses static conservation laws in the former and square root flux continuity/dynamical probability in the latter.
We have suggested in previous notes that there is a tendency to consider quantum mechanics as a kind of hybrid probabilistic-deterministic theory because it is supposed to ultimately “merge’ with classical mechanics for high energy levels (tiny wavelengths). Newtonian mechanics is deterministic and so somehow it seems quantum mechanics should also be seen as deterministic, hence the sense of surprise at tunneling when KEave(x)=0, even though vrms(x)=0 does not mean that actual values creating the probability are 0. Here, we wish to argue that exp(-iEt+ipx) (or the exp(ipx) part for spatial interactions) is really a momentum probability scheme which may possibly bypass many aspects of the actual physics to describe a problem in a simpler form in terms of a momentum conserving probability. A key idea is that exp(ipx) is actually the form to use when one has randomness as we will argue for reflection-refraction from an n1-n2 index of refraction junction. We argue that this idea is already known for a particle in an infinite well for which one has sin(pave x), i.e. 1/2i (exp(i pave x) - exp(- i pave x)). In other words, one has the form exp(i pave x), even though it is linked with an average p. Given that one is not considering deterministic physics, there is nothing wrong with exp(i pave x) being composed of Sum over p a(p) exp(ipx), where exp(ipx) holds in all space, not just within the infinite well potential. exp(ipx) is a momentum conserving probability, but on average momentum must also be conserved by a free statistical entity within the well. Similarly, W(x) for a single particle bound state consists of many exp(ipx)s, but represents a fixed E ave and on average this Eave must be conserved in time, hence the time dependent exp(-i Eave t). As a result, one may see that the quantum mechanical exp(-iEt+ipx) is an attempt to describe a system in a simple manner in terms of momentum conserving probability. To see this in more detail, we consider (1) which describes reflection and refraction of a photon at an n1-n2 junction in depth. In a previous note, we have already pointed out that in n2, c/n2 simply means that the photon is interacting on average with atoms/molecules in n2, but has c in between such interactions. It is only on average that one has c/n2 and even E and p2 = p n2. Nevertheless, one may use exp(ip2 x) as a probability which describes average properties of the system, just as it does in the examples in the above paragraph. The question then becomes: Do the microscopic details of the photon physics (the actual interactions) dictate the probabilities to reflect/refract or does the simplified exp(ipx) set linked to momentum conservation do so? Based on (1), it seems that reflection is completely independent physically from “refraction”, the latter requiring re-radiation and affecting all glass molecules. As a result, it appears that it is difficult to suggest that it is the microscopic interactions which dictate probability because reflection and refraction are microscopically so different. On the other hand, reflection-refraction is not a .5-.5 probability random process as experimental results show. The question then becomes: If some kind of systematic probability driver is not supplied by the microscopic interaction, what provides it? We argued in previous notes that one may introduce the probability exp(ipx) into Newtonian scattering to account for an initial p1,p2 (momentum vector set) having equal probability to produce any pi, pj (outcome set) as long as momentum is conserved. We suggest here that it is exp(ipx) that one must use for random interaction if one insists on conserving momentum. Thus, exp(ipx) bypasses microscopic details of interaction and provides an “ any outcome pair of momenta” is equally likely if momentum is conserved. This is a major simplification of the problem, but the question is: What constraints does it impose on physical results? For a free particle, it suggests that it may interact probabilistically with both slits of 2-slit apparatus if the slits are about hbar/p apart. For randomness, one would expect equal probability for each slit and this is the case as long as one also considers exp(ipx) for each slit to a the same point on a screen. In the case of reflection-refraction at an n1-n2 index of refraction junction, we try to see how the simplification exp(ipx) can be consistent with the microscopic details of the interactions and at the same time dictate the probabilities to reflect and refract. We try to consider why an exp(ipx) scheme predicts how nature behaves while a random one with no momentum conservation considerations does not. In particular, we argue that exp(ipx) represent randomness in a two body scattering problem in which momentum is conserved and so this same exp(ipx) must be carried over into problems which involve interactions instead of simply suggesting that one has .5 probability to reflect and .5 to refract because the two processes are independent and hence random. They are random, but with conserved momentum and hence an exp(ipx) scheme is required, we argue.
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 (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.