In (1), it is argued that the Schrodinger cat paradox arises due to a superposition state being considered “physical” instead of an “an expectation over possible future outcomes” . This is consistent with ideas linked to a quantum directional or interaction probability versus a state probability which we have discussed in various notes (2). In particular, classically there exist states, like heads up for a coin, a certain number of a die, an incident, reflected or refracted state for a photon, etc. We have suggested in (2) that exp(ipx) is a directional or interaction probability in that this probability is linked to the probability to have a p in a two-body scattering experiment and is involved in conservation of momentum. A state probability should not depend on x as the state is determined by p (and mo). For example, an incident, reflected or refracted photon probability does not depend on x, rather it is specifically given by AA/c, BB/c or CC/c2, where c is the speed of light in n1=1 (index of refraction), c2= c/n2 and AA,BB,CC are the fluxes of the incident, reflected and refracted photons. Thus, exp(ipx) is linked to the state through p, but through x it is linked to position. This means that for a given p there is uncertainty in x quantified by the wavelength hbar/p. This means there is position uncertainty with respect to an impulse hit making this an interaction probability and a directional one because p and -p have different effects for a conservation of momentum equation. As argued in (2), interactions are based on exp(ipx) because they involve impulse hits and so even though classical state probabilities exist, the physical problem is solved through considerations of exp(ipx)s. In particular, sums or OR situations of exp(ipx)s arise due to uncertainty in x. It is the directional/interaction probability exp(ipx) which predicts the x uncertainty and also appears in so-called superposition equations which represent different possible outcomes which may all exist in a tiny dx during the interaction. One should not think in terms of a state vector, but rather possible interaction probabilities exp(ipx). These are OR or sums of possible probabilities which interact in x and not a mixture of states. In other words, one does not have a dead cat mixed with a living one or an incident photon mixed with a reflected one. Rather, one adds directional/interaction probabilities for different states because one is uncertain which state exists in dx. This leads to math conditions on the exp(ipx)s as discussed in (2) which then ultimately govern the classical probabilities for one to have a particular state or another. The point we make is that exp(ipx) is not a state probability, but an interaction one which is consistent with the ideas of (1).
In (1), it is argued that the Schrodinger cat paradox arises due to a superposition state being considered “physical” instead of an “an expectation over possible future outcomes” . This is consistent with ideas linked to a quantum directional or interaction probability versus a state probability which we have discussed in various notes (2). In particular, classically there exist states, like heads up for a coin, a certain number of a die, an incident, reflected or refracted state for a photon, etc. We have suggested in (2) that exp(ipx) is a directional or interaction probability in that this probability is linked to the probability to have a p in a two-body scattering experiment and is involved in conservation of momentum. A state probability should not depend on x as the state is determined by p (and mo). For example, an incident, reflected or refracted photon probability does not depend on x, rather it is specifically given by AA/c, BB/c or CC/c2, where c is the speed of light in n1=1 (index of refraction), c2= c/n2 and AA,BB,CC are the fluxes of the incident, reflected and refracted photons. Thus, exp(ipx) is linked to the state through p, but through x it is linked to position. This means that for a given p there is uncertainty in x quantified by the wavelength hbar/p. This means there is position uncertainty with respect to an impulse hit making this an interaction probability and a directional one because p and -p have different effects for a conservation of momentum equation. As argued in (2), interactions are based on exp(ipx) because they involve impulse hits and so even though classical state probabilities exist, the physical problem is solved through considerations of exp(ipx)s. In particular, sums or OR situations of exp(ipx)s arise due to uncertainty in x. It is the directional/interaction probability exp(ipx) which predicts the x uncertainty and also appears in so-called superposition equations which represent different possible outcomes which may all exist in a tiny dx during the interaction. One should not think in terms of a state vector, but rather possible interaction probabilities exp(ipx). These are OR or sums of possible probabilities which interact in x and not a mixture of states. In other words, one does not have a dead cat mixed with a living one or an incident photon mixed with a reflected one. Rather, one adds directional/interaction probabilities for different states because one is uncertain which state exists in dx. This leads to math conditions on the exp(ipx)s as discussed in (2) which then ultimately govern the classical probabilities for one to have a particular state or another. The point we make is that exp(ipx) is not a state probability, but an interaction one which is consistent with the ideas of (1).
Classically, probability is considered in terms of the number of possible outcomes. For example, a die toss has six possible outcomes and a coin toss, two. Probability calculations make use of probabilities and hence of the total number of possible outcomes. Thus, they are not deterministic calculations which follow “what happens” as in Newtonian mechanics. We suggest that if free particle quantum mechanics is probabilistic, it must also involve the notion of possible outcomes in an experiment and not represent a deterministic approach. We suggest, however, that something interesting applies to quantum mechanics because we have suggested that free particle quantum mechanics arises from the Lorentz invariant A = -Et+px. We note that there exist x,t trajectory points which follow x=vt. We also note, however, that one may introduce the idea of intervals dx, dt (because Newton’s dx→0, dt→0 is an idealization) and that A is unchanged for x,t and x+hbar/p , t+hbar/E. In other words, we suggest that there are intervals with probability about each x,t trajectory point. A trajectory point is associated with deterministic physics, but probability about this point is not. In other words, one has a kind of hybrid probability scheme. Thus, exp(ipx) and exp(-iEt) really represent fluctuation probabilities about a trajectory point x and t. These ranges of possible x and t fluctuations about a trajectory point x,t allow for further probability to occur in certain interactions. We consider two examples. The first is 2-slit interference which involves a deterministic particle moving along a straight line towards a 2-slit mechanism perpendicular to its path. At the 2-slit apparatus, one considers a probabilistic calculation as the uncertainty in x, i.e. dx=hbar/p allows for possible interaction with both slits. This involves considering all possible outcomes (i.e. motion through one slit or the other which is a classical situation), but exp(i p dot r1) and exp(i p dot r2) “sit on top” of these trajectories at the 2-slit apparatus and one has interference of the probabilities linked with each classical possible outcome. Thus, there are two sets of probabilities which are intertwined, the probability range hbar/p for each path and the two possible paths which have equal probability. One must consider the possible outcomes for both sets together. We suggest that a similar analysis applies to the case of 1-D reflection-refraction for x=0 at an n1-n2 index of refraction junction. Thus, we suggest that a deterministic free particle with x=vt may encounter probabilistic interaction situations, such as a two-slit apparatus or an n1-n2 index of refraction junction. These are probabilistic only because there is a probability fluctuation scheme in x associated with each deterministic x point. It is this probability which allows for a second probability to arise due to interaction with the slits or at the 1-D n1-n2 junction. In such a case, one must consider all possible outcomes which means that one does not only consider the exp(ipx) type probabilities, but also those linked with the physical interaction, i.e. two possible paths, one from each slit, or an incident, reflected and refracted photon at x=0 (n1-n2 junction). It is a combination of all possible paths which must be used in calculations. We thus suggest that even though there is a sense of determinism, i..e free particles move as x=vt and these exist in the 2-slit case and the 1-D reflection-refraction case, in a probabilistic calculation one must consider all probabilistic outcomes consistent with these. Thus, quantum mechanical calculations are true probabilistic calculations in the sense that they involve all possible outcomes in an analysis.
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.
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.
It is known that a single photon moving in the x direction and encountering an n1-n2 index of refraction discontinuity at xo will either reflect or refract, with the probabilities for each summing to 1. This problem, however, is solved using a balance of Aexp(ipx)+Bexp(-ipx) = Cexp(ip2 x) at xo and the derivative d/dx, of this equation. The complex conjugate of one equation multiplied by the second yields: AA/c = BB/c + CC/c2 where E=pc and E is constant for the incident, reflected and refracted photon. The two continuity equations apply even in the case of a single photon, but in such a case one has either an incident photon changing to a reflected one or an incident to a refracted one. One never has all three together as in thte two continuity equations. We ask: How is it possible to include both a refracted and reflected photon together with the incident in the same equation? We suggest here that the form exp(ipx) applies to momentum, whether it is part of a particle or part of a potential field. exp(ipx) may be obtained from maximization of entropy -P(x)ln(P(x)) subject to the periodic constraint ipxP(x). We suggest that in a single photon interaction, one has momentum in media n1 and n2 and these allow for the continuity equation with the system interactions taking the place of a photon to create the balance. There is stochasticity in that a single photon either becomes a reflected or refracted one, but exp(ipx) does not simply apply to the photon. it may be applied to the n1-n2 interface momentum which is created during the interaction. This interaction is consistent with the a steady stream picture only involving photons . Thus for a single photon, it appears as if one has an entangled probability exp(ipx)’s for the incident, reflected and refracted photons, but actually the physical photon only plays a role as first an incident one and then either a reflected or refracted one with the n1-n2 interface momentum creating the missing exp(ipx) piece needed for the continuity balance. Ultimately, A,B and C become probabilities AA/c,= 1, BB/c = P(reflected) and CC/c2= P(refracted), but these are not obtained by maximizing entropy subject to a constraint.as one does not have independence of probabilities.We note that the two continuity equations combine to create a pressure balance which one wishes to have physically in the problem. The photon reflection-refraction problem may be reformulated for a particle with rest mass using a particle with E, p in a constant potential V1 reaching a V1-V2 (both constant) interface at xo with E remaining constant.
It is sometimes stated in the literature (1) that quantum mechanics is based on conditional probabilities. If this is the case, one would expect such conditional probabilities to appear in a single particle bound state. A free quantum particle is associated with a probability to interact of exp(ipx) (linked to 2-slit interference, 1-D reflection-refraction at an n1-n2 junction etc), but the kinetic energy and momentum are pp/2m (nonrelativistic) and p at all x even though each x has a different phase according to exp(ipx). The momentum is p at x, but the probability for an impulse hit at x is not the same even though its modulus is. When different probabilities exp(ipx) add in an OR situation, this leads to a spatial pattern which no longer has a modulus of 1, i.e. translational invariance is broken. Such a pattern also occurs for a single particle bound state W(x)=Sum over p a(p)exp(ipx). W(x) can have 0 points, but the average kinetic energy -1/2m d/dx dW/dx / W is not 0 at such a point. One has, on average (a statistical calculation), no average interaction at such an x point. One might ask why this should have anything to do with average kinetic energy? If one writes a conservation law: KEave(x) + V(x) = 0, then one considers kinetic energy interacting with potential energy. If there is no interaction at some x point for a given p (not W(x) on average), then it should not be part of the conservation of energy at this point. One is only interested in kinetic energy which interacts with potential energy at each x point. Thus, a(p) exp(ipx) + a(-p) exp(-ipx) (with a(p)=a(-p)) may have a 0 point at some x and so it does not have an average interaction at such a point, i.e. pp/2m (a(p)exp(ipx) + a(-p)exp(-ipx)) = 0. As a result, in a bound state, one considers kinetic energy per p, -p multiplied by the average probability (p, -p). Different a(p)exp(ipx) + a(-p)exp(-ipx) (a(p)=a(-p)) weights in this average picture indicate the extent to which the particle interacts. This idea holds for a single p,-p pair, but one must normalize over all such pairs, leading to the idea of a conditional probability. Given that hbar/p must be “created physically” in space so to speak, one must have interaction probability which creates this physical situation even in cos(px)+ i sin(px). This leads to the idea of interaction probability being 0, positive, but <1, or even negative and it is the interaction which allows one to consider conservation of energy with a potential V(x) in the first place. As a result, one cannot include a full pp/2m if there is negative, 0 or only particle interaction. Thus, pp/2m is multiplied by 2 a(p)cos(px) /W(x) for a(p)=a(-p).
In Part 1, we argued that the consideration of all possible outcomes is a defining idea of a probabilistic calculation. We then argued that in free particle quantum mechanics there were two sets of outcomes, one driving the other. In particular, a single particle moves as x=vt, but at each x,t point, one has interaction uncertainty/fluctuations of dt and dx linked with exp(-iEt) and exp(ipx). As a result, in a 2-slit interference problem, exp(ipx) (a probability linked with possible positional fluctuations about x) leads to a further probability of interacting with both slits. The two sets of outcomes must be considered, i.e. exp(i p dot r1) + exp(i p dot r2), where r1 is a vector from the center of the first slit to a point on a screen far away, and r2, a vector from the center of the second slit to the same screen point. We also considered the case of reflection-refraction from an n1-n2 index of refraction junction. The point is that these two cases are clearly probabilistic, i.e. not deterministic at all. Here we consider two different scenarios which have the appearance of a deterministic result, but are actually probabilistic and so linked with all possible outcomes, i.e. an average over these which must be understood as an average (as in classical probability). The first is a single particle bound state and the second, transmission through a potential wall. In the bound state case, the analogue is a Newtonian bound state and in fact between the classical turning points KEave(x) + V(x) = E in the probabilistic case which exactly matches the classical one. One is, however, dealing with an average KEave(x) = -1/2m d/dx d/dx W/ W, where W=Sum over p a(p)exp(ipx), it involves all possible outcomes and is not a physical state. Given the similarity to a deterministic problem there is a danger of not considering the quantum bound calculation as a probabilistic one. In the quantum single particle case, one may be surprised by tunneling, but the quantum probabilistic scenario is one in which all possible interaction outcomes are considered together (as is required by probability) and so one cannot possibly ignore interactions linked to a particle entering or leaving the potential region. Furthermore, it is only KEave(x) which equals the classical result and so E which is single value in the classical case, is linked to an average value in the quantum case, i.e. a particle may have a p such that pp/2m is not zero at the classical turning point. Only KEave(x) =0 at this point. An interesting feature is that a probabilistic calculation applies to a single particle run (which only produces one outcome). One, however, cannot measure the full probability associated with a problem without making a very large number of measurements.