In classical statistical mechanics, although one has averages of a system, a single particle within the system has properties which are sharply defined at any x,t, even though these properties change stochastically, i.e. through interactions. A quantum free particle, on the other hand, has information based on the Lorentz invariant -Et+px, multiplied by i to create periodicity which indicates an inherent information, not one based on the outcome of trial runs or interactions. The probability is exp(-iEt+ipx). If E and p are considered as constant, then there are independent probabilistic fluctuations in x and t, which means there may be corresponding probabilistic interactions, both impulse ones (one dimensional reflection/refraction, n-slits) and temporal ones (Rabi frequency). In such a case, one has interference and the form exp(ipx) or exp(-iEt) governing the statistics, not a maximization of entropy linked to interaction/outcome results. Nevertheless, the quantum particle interacts with fields, it seems, which give rise to p or E changes. It seems the key idea is that one has stochastic uncertainty before an interaction occurs and hence interference. The alternative is that such stochastic uncertainty does not exist before a specific interaction, i.e. the inherent probability exp(-iEt) or exp(ipx) has no effect on the interaction. This is not to say that there cannot be probabilities before the interaction. In a Maxwell-Boltzmann gas, there are probabilities P(e1) and P(e2) for particles with e1 and e2, but a specific e1 and e2 have no uncertainty. The uncertainty is in the outcome of the reaction. Another example is a two state electron in an atom which absorbs and emits photons. The rules for time evolution are that the ground state is represented by exp(-iEot), but at some stochastic t value, it changes to exp(-iE1t) and vice versa. The ground state absorbs a photon and the excited emits. The two states are clearly defined by the differing interactions which govern their time dependence which is taken to be simply the changing of Eo to E1 and vice versa. This problem is then based on probabilities based on outcomes of an interaction and maximization of entropy subject to a constraint is employed. There is no initial interference.
In classical statistical mechanics, although one has averages of a system, a single particle within the system has properties which are sharply defined at any x,t, even though these properties change stochastically, i.e. through interactions. A quantum free particle, on the other hand, has information based on the Lorentz invariant -Et+px, multiplied by i to create periodicity which indicates an inherent information, not one based on the outcome of trial runs or interactions. The probability is exp(-iEt+ipx). If E and p are considered as constant, then there are independent probabilistic fluctuations in x and t, which means there may be corresponding probabilistic interactions, both impulse ones (one dimensional reflection/refraction, n-slits) and temporal ones (Rabi frequency). In such a case, one has interference and the form exp(ipx) or exp(-iEt) governing the statistics, not a maximization of entropy linked to interaction/outcome results. Nevertheless, the quantum particle interacts with fields, it seems, which give rise to p or E changes. It seems the key idea is that one has stochastic uncertainty before an interaction occurs and hence interference. The alternative is that such stochastic uncertainty does not exist before a specific interaction, i.e. the inherent probability exp(-iEt) or exp(ipx) has no effect on the interaction. This is not to say that there cannot be probabilities before the interaction. In a Maxwell-Boltzmann gas, there are probabilities P(e1) and P(e2) for particles with e1 and e2, but a specific e1 and e2 have no uncertainty. The uncertainty is in the outcome of the reaction. Another example is a two state electron in an atom which absorbs and emits photons. The rules for time evolution are that the ground state is represented by exp(-iEot), but at some stochastic t value, it changes to exp(-iE1t) and vice versa. The ground state absorbs a photon and the excited emits. The two states are clearly defined by the differing interactions which govern their time dependence which is taken to be simply the changing of Eo to E1 and vice versa. This problem is then based on probabilities based on outcomes of an interaction and maximization of entropy subject to a constraint is employed. There is no initial interference.
In (1), it is stated that “time flows continuously like a river’ … and is treated like a classical variable in Quantum Mechanics.” Here, we wish to consider the behaviour of time in quantum situations. We argue that a 4-space sense of time appears in quantum mechanics through exp(-iEt+ipx), but only intertwined with E (energy) and p (momentum), i.e. interactions involving these variables. Furthermore, there suggest that there must be a clear measure of t as well as an interaction changing E to use exp(-iEt) and a clear measure of x and an interaction changing p to use exp(ipx). In particular, we note that the classical equation x/t = v ensures that time and x are not on the same footing and this same equation holds in special relativity, so -Et+px = constant ((1)) which has the appearance of x and t being on the same footing (as well as (x,ct)) is subject to x/t=v (for xo=0, to=0). Thus, if x/t=v governs x and t information, then x and t are not independent. It seems that one would require fluctuations from these values. The fluctuations ranges could be linked, but actual t and values within the fluctuations would be independent. . We suggest it is possible to find a formalism which treats x and t independently from ((1)), but this formalism must be intertwined with E and p. In particular, one has free particle quantum mechanics in which delta t = hbar/E and delta x = hbar/p. X and t are decoupled in this scenario and so appear as independent because there is no notion of x/t=v when dealing with quantum type interactions. When dealing with such interactions, one uses the probability exp(-iEt+ipx), but this begs the question: How is this factor really used? We suggest that one must consider the following. Given exp(ipx), one uses it if p changes and there is a distinct measure of changing x in the problem. This scenario applies to two-slit interference, bound states with V(x) and one dimensional reflection-refraction at a point x=0. Given exp(-iEt), one uses it if there is a distinct interaction involving E, i.e. it changes and there is a distinct measure in time. Thus, two-slit interference, one dimensional reflection-refraction and elastic scattering from a V(x) are out because they don’t involve changes in E. A bound state with V(x) certainly involves changes in E, but does not have a direct measure of time t (independent of x). A bound state with V(x,t), however, does have a direct measure of t (being independent of x) and also involves energy changing and so exp(-iEt) may be used in this case, as will be shown.
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.
In previous notes (1), we argued that stochasticity (in x and t for a fixed E energy and p momentum) may occur at a single particle level based on the Lorentz invariant A = -Et+px ((1)). This invariant is also the relativistic/nonrelativistic classical action for v=x/t. The argument made in (1) was that a fluctuation in t of constant/E could cancel one in x of constant/p. We further argued one could create eigenvalue equations using id/dt and -id/dx and obtain the two dimensional probabilities exp(-iEt) and exp(ipx), i.e obtain a description of free particle quantum mechanics. Experimentally, both exp(-iEt) and exp(ipx) have been verified through interference patterns. Theoretically, however, ((1)) was known about 1905 and no single particle stochasticity was associated with it. In this note, we try to argue why there should be an associated stochasticity associated with ((1)) based on motion through a constant potential for a particle with rest mass and motion through a medium with an index of refraction for a photon. We in fact argue that one cannot solve certain photon problems such as one dimensional reflection/refraction without explicitly considering the stochasticity which is usually left out when describing p2=pn2 and c2=c/n2 where n2 is the index of refraction in the second medium and p and c are values in the first medium with n1=1. We further consider exp(ipx) and exp(-iEt) as arising from the maximization of Shannon’s entropy subject to certain constraints. The product of these two two-dimensional periodic probabilities is exp(-iEt+ipx) which we argue must hold as seen from observers moving with constant velocity. Thus we argue that the Lorentz invariant -Et+px follows from statistical considerations alone. Given this expression (a subset of the whole picture), one may try to create a deterministic viewpoint by setting x/t=v=constant. This allows one to consider transformations which define E and p in terms of mo and v and yields the invariant EE = pp+momo (c=1). Thus we argue that the conservation of energy equation ultimately follows from a deterministic view of the argument of a statistical probability. In other words, it is only a subset of the full picture i.e the quantum mechanical statistical one.
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 previous notes, we argued that one may find the source of exp(-iEt+ip dot r) in Newtonian mechanics by considering two body elastic scattering. Even though Newtonian mechanics is deterministic, probability is routinely introduced in such elastic scattering events because one cannot deterministically predict the outcome in most cases. Rather, one relies on conservation of energy and momentum and any possible conforming solution has equal probability. We argued that a free particle does not have a real valued probability weight because each is independent of the other (unlike in an ideal gas). As a result, we suggested the probabilities exp(iE C1) and exp(ip C2), with the constants C1 and C2 accounting for units. We then went on to make arguments linked to time reversal symmetry for exp(ipC2) and overall Lorentz invariance to finally arrive at exp(-iEt+ip dot r). Here, we wish to examine these considerations in more detail as we think there is a more general principle present, namely that of a state probability. We first note that even though exp(iEC1) and exp(ipC2) are consistent with conservation of energy and momentum, i.e. exp(iC1 E1) exp(iC1 E2) = exp(iC1 E3) exp(iC1 E4) and a similar expression for p, one only requires such conservation if all particles are classically at the same x and t, i.e. the collision point. As a result, simply given exp(iC1 E1) and exp(iC1 E2), the particles with E1 and E2 may be far apart in space. As a result, it is not clear why one would create a product probability with them unless they were actually interacting at the same x,t classically, but mathematically there is nothing preventing this and claiming conservation even though there is no reaction. It is this idea which suggests that exp(iC1 E) and exp(iC2 p) should be linked with x and t.Then, conservation only occurs in products if x and t are the same for different particles, otherwise it does not follow. This might suggest: exp(i E * (f(x)+g(t)) ) and exp(i p * (f(x)+g(t)). Here p is taken as a one-dimensional number along the direction of momentum. It may be positive or negative.f and g are unknown functions. We next note that x is measured from an origin. Thus, x1 and x2 are distances measured from an origin as is x1+x2. If one writes exp(i E * (f(x)+g(t)), this seems to imply that x and t are both measured from zero points. If one multiplies by exp(i E * (f(x2)+g(x2)), it seems one would wish this result to be equivalent to exp(i E (f(x+x2)+g(t+t2)). This leads to f and g being linear functions in x and t. Now one may have noticed earlier that including E,p,x and t in the “conservation” probability for a free particle means that one has completely defined the state of the particle. This probability is a state probability and so even though it is complex, one may argue that this is a property which should not change through a Lorentz transformation, just like particle number. This then leads to the notion of exp(-iEt+ip dot r). Thus, one initially attempts to describe the probability to have a particular p and E such that conservation of momentum and energy holds. In such a case, there is no need to introduce x and t and a Lorentz transformation changes the probabilities, but not their conservation properties as is shown. Thus, it is the desire to enforce conservation of energy and momentum only when a classical reaction occurs with all particles at the same x and t which introduces x and t into the probability along with E and p making it a state probability which should be Lorentz invariant. We suggest that there is a further important consequence of introducing a state probability. Given that it does not change under a Lorentz boost, it is like a quantity and may be used in OR situations a priori, even though it does not represent an obvious physically conserved quantity. This allows one to immediately write: Aexp(ipx)+Bexp(-ipx)=Cexp(ip2x) at x=0 for 1-D reflection-refraction from an n1-n2 index of refraction junction. (One may write a second equation involving p as exp(ipx) represents the probability to have p at x.) This then leads to a quantum mechanical manner of dealing with an interacting free particle.
In Part III we considered the quantum free particle wavefunction exp(ipx) as consisting of two parts cos(px) and sin(px) treated as two different dimensions. We also argued that each, is a probability distribution in space and so associated with an internal force. One may notice that exp(ipx) does not contain time, but still predicts the direction of motion. Given that one cannot consider x(t) within a wavelength hbar/p even though on average the particle moves as x=p/m t, there is no real sense of time within a wavelength. Extending this idea, one repeats cos(px) humps. Given that the particle is moving, one must somehow juxtapose two spatial probability densities. The derivative of the distribution is proportional to average momentum and so one may see the notion of action-reaction occurring between cos(px) and sin(px) and within cos(px) and sin(px) accounting for the sign changes with x. We argue that d/dx cos(px) regions which change sign imply that the impulse direction changes. Furthermore, cos(px) has regions near the endpoints of the wavelength which have very low values (close to zero) which is fine if one has two juxtapositioned distributions. If, however, one creates a single particle bound state, exp(ipx) and exp(-ipx) for a(p)=a(-p) (a specific parity) give the appearance of a cancelation leaving the impression of 2cos(px) i.e a single “static” distribution. This occurs because there is no internal time in the picture and so p and -p probabilities add. The so-called canceled pieces have not really disappeared. as will be argued, they simply do not contribute to an average impulse which is what experiments apparently detect. Thus “negative” quantum probability is linked to impulse in the opposite direction, we argue. Thus interference is really linked to the notion of probabilistic impulses i.e. p exp(ipx) even though exp(ipx) is used as a kind of probability. exp(ipx) contains internal action-reaction impulse hits which have experimental consequences as we argue is shown in a reflection/refraction problem, but are also linked with impulse hits associated with p. Thus there are two sets of forces to deal with. This, we argue, explains why one writes the energy conservation equation AA/c1=BB/c1+CC/c2 for reflection/refraction where A,B,C represent the incident, reflected and refracted rays and AA simply shows that intensity/flux is positive as two equations: A+B=C and Ap - pB = p2C with E=pc. The two equations,, needed to solve for B,C in terms of A,, describe the internal force in exp(ipx) and the probabilistic impulse p exp(ipx). For a single particle bound state, one considers a linear combination of free particle states indicating that impulse hits represent the underlying physics. Given the apparent spatial invariance of a free particle exp(ipx), one may wonder how a kinetic energy profile appears. We argue that different p values yield different peak points for maximum impulse hits and these are out of synch leading to a spatial distribution of KE(x) or d/dx KE(x) which is equivalent to -dV/dx =Force where V(x) is the potential.