Quantum mechanics seems to function at two levels in terms of probability. One level is represented by classical probabilities P(x)=W*(x)W(x) and P(p)=a*(p)a(p) where W(x)=wavefunction= Sum over p a(p)exp(ipx). For a free particle this level maps into continuous space and time i.e. exp(-iEt+ipx)*exp(-iEt+ipx)=1, but at a more fundamental level (the wavefunction level) there seems to be spatial and time resolution. Spatial resolution for a free particle is based on exp(ipx) i.e. a resolution pattern cos(px) and its space shifted sin(px) representing motion. This resolution carries over to bound states for which P(x) is not 1, but is represented by crests and troughs. exp(-iEt) represents resolution in time, we argue and is linked to 1/E where E is an energy associated with an interaction. As a result a bound system may have an overall exp(-i (M1 + m2 + En) t) where En is binding energy and an overall frequency (or spatial resolution of 1/(M1 + m2 + En), but exp(-iEnt) represents a frequency of 1/En which is linked to the interaction between M1 and m2. Thus the time resolution seems to be associated with energy and there may be various kinds of energy in a system (e.g. rest mass energy, binding energy) etc. These appear to be linked to different time resolutions. If En becomes 0, i.e. M1 and m2 no longer interact, then M1 has a time resolution proportional to 1/M1 and m2, to 1/m2. Thus at the wavefunction level it does not seem there is an absolute time, rather there are various time resolutions associated with different energies.
Quantum mechanics seems to function at two levels in terms of probability. One level is represented by classical probabilities P(x)=W*(x)W(x) and P(p)=a*(p)a(p) where W(x)=wavefunction= Sum over p a(p)exp(ipx). For a free particle this level maps into continuous space and time i.e. exp(-iEt+ipx)*exp(-iEt+ipx)=1, but at a more fundamental level (the wavefunction level) there seems to be spatial and time resolution. Spatial resolution for a free particle is based on exp(ipx) i.e. a resolution pattern cos(px) and its space shifted sin(px) representing motion. This resolution carries over to bound states for which P(x) is not 1, but is represented by crests and troughs. exp(-iEt) represents resolution in time, we argue and is linked to 1/E where E is an energy associated with an interaction. As a result a bound system may have an overall exp(-i (M1 + m2 + En) t) where En is binding energy and an overall frequency (or spatial resolution of 1/(M1 + m2 + En), but exp(-iEnt) represents a frequency of 1/En which is linked to the interaction between M1 and m2. Thus the time resolution seems to be associated with energy and there may be various kinds of energy in a system (e.g. rest mass energy, binding energy) etc. These appear to be linked to different time resolutions. If En becomes 0, i.e. M1 and m2 no longer interact, then M1 has a time resolution proportional to 1/M1 and m2, to 1/m2. Thus at the wavefunction level it does not seem there is an absolute time, rather there are various time resolutions associated with different energies.
The wavefunction for a quantum free particle is exp(-iEt + ipx) In this case, E and p are linked through either E=pp/2m (nonrelativistic case) or EE =pp + momo (relativistic case c=1). We argue that there are different interactions involved with E and p. P (momentum) is energy flow and is involved in impulse type interactions with a potential usually V(x). For V(x,t) matters are more complicated, but will also be discussed. The impulse hits are associated with exp(ipx) and lead to interfering plane waves and an overall crest-trough spatial scenario. There is a kinetic energy (nonrelativistic case) associated with each p, but a new overall energy En (n= eigenlevel) is created which yields the free form exp(-iEnt). This energy is linked to the cycling of exp(ipx)’s. It is also part of the new mass of the bound state system i.e. mo+En. This new system could be considered as an overall free particle and obtain a new overall momentum. (The initial bound state has 0 overall momentum.) Thus, time and space parts of the wavefunction behave in very different manners due to the nature of the interaction even though there is a link between 1/E and cycling of exp(ipx) terms. We call this case an impulse or space based type of interaction. It is also possible to have a time based interaction i.e. the absorption of a photon, phonon etc. In such a case, there is momentum conservation and so the whole system receives a new momentum, but this is usually ignored. The main effect is for the system to move from one bound state E(n1) to another E(n2). This has consequences on the spatial density and cycling of exp(ipx) terms in the wavefunction, but is not an impulse interaction in space. In the case of a system: W(x,t)= Sum over n an(En) exp(-iEnt) Wn(x), one has time based interactions which knock the system from one bound state to another (as in heat bath interaction). Again there are links between each En and the spatial density, but there is an overall spatial density(x,t) and average energy(x,t) which depends on the interference of exp(-iEnt) terms. (This disappears if one integrates over all x.) This is similar to the diffraction/interference spatial pattern, but in time. In the impulse scenario for a(p)=a(-p) one has cross terms in the spatial density of cos( (pi-pj)x) which create the crest/trough pattern seen. For the heat bath time type interaction, there exist cos( (Ei-Ej)t) type terms if an(En)=a*n(En) which are linked to the energy Ei-Ej of the intermediary photon/phonon etc and its time period. Thus in both cases, the quantum crest/trough pattern (in space or time) seems to be linked to the momentum or energy transfer. The energy transfer is known to be linked to periodic objects (photons/phonons) and it seems the momentum transfer pi-pj also has a similar periodicity, but in space. It is almost as if these “entities” are creating the unusual quantum crest/trough patterns.
Free quantum particles may be represented by a wavefunction of exp(ipx) where “px” suggests a resolution scale L with p=2(3.14)hbar/L. In this note, we suggest that such a resolution may already be present in classical mechanics. exp(ipx) is a periodic form which incorporates this resolution and through a modulus of 1 ensures equal probability to be at any point x. In order to use exp(ipx), it seems that there must be a physical mechanics associated with the resolution. Given that exp(ipx)=cos(px) + isin(px), we argue that two different probability cycling or flows exist, one related to energy (pp/2m) and time and the other to flux- momentum and space i.e. sin(px). The link between the two is: cos(px)cos(px)+sin(px)sin(px) =1 i.e. the two are out of phase. This appears to be similar to the two dimensions x and ict. Each cycle or flow has its own frequency with positive and negative mutually exclusive subdivisions within resolution pockets, 1/E for time and L with p=2(3.14)hbar/L for space. The frequencies are associated by positive and negative probabilities (linked perhaps with forward and backward motions etc) and lead to interference if one considers different momentum values. We attmept to consider these issues in this note. In addition, we examine ln(W) representing information, with W(x), being the wavefunction . The change in information is -i dx . Thus, it seems that momentum flow carries information in a quantum bound state.
In classical physics, a potential V(x) acts on a particle at a location x regardless of its velocity which is taken to be constant within dx (even though the particle accelerates). Thus writing potential energy V(x) at a given x is like taking a snapshot of the particle when it is at x. The snapshot is static and does not indicate the dynamics of the problem as the particle may be moving either to the right or left, which is irrelevant in terms of the exact potential energy and even kinetic energy at point x. A quantum particle moving with constant velocity is described by a two-dimensional dynamic probability exp(ipx), with cos(px) representing a static density and sin(px) a time-shifted density to capture one directional motion such that exp(ipx)exp(-ipx)=1, we argue. V(x), however, only acts on the static density (linked to cos(px) and not sin(px)). Thus it seems that bound state quantum mechanics needs to remove the dynamic pieces of free particle wavefunctions exp(ipx) through interference. One might argue that for wavelengths of the order of the system length one does not know if one has exp(ipx) or exp(-ipx) at an x point, but for high energy levels, wavelengths become small with respect to the system length and yet interference is still present. Thus it seems that for the calculation of energy one requires a static average snapshot of the bound particle. Physically, we argue that the dynamic piece is still present i.e.. if one knocks out a bound particle and it is found in the p momentum state, it is represented by exp(ipx). Thus the purpose of the bound state wavefunction is to facilitate the calculation of average energy which is associated with a static snapshot type of density and not necessarily represent all of the dynamical information in the system i.e. the two dimensionality of each exp(ipx). As a result, the bound wavefunction does not show motion to the right and then to the left in the high energy limit.
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 a previous note (1) we considered a free particle entropy based on the real part of the wavefunction exp(ipx) i.e. |cos(px)|. |cos(px|) may be normalized over a certain length, just as exp(ipx) is. The idea is that there exists an uncertainty in space within a wavelength which should be reflected in an entropy expression. As p becomes very large, cos(px) becomes very narrow and has many peaks in L. Thus probability →0 so p ln(p)--> 0 using l’Hopital’s rule. This coincides with a free classical particle having zero entropy. In this note, we revisit arguments presented previously, suggesting that A= -Et+px, a Lorentz invariant and also the free particle classical relativistic and nonrelativistic action for v=x/t We suggested in the past that A is responsible for both the wavelength hbar/p and the time period hbar/E. Here we suggest uncertainty within a time period should be considered together with uncertainty within a wavelength. For instance, recent experiments (2) have shown that one may have quantum interference in time as well as space. Thus there is uncertainty for a free particle in both space and time. We add the entropies for these two. Next, we consider the case of a bound system. Such a system creates a crest-trough pattern (with different heights and widths) for its spatial wavefunction, but E, average energy is constant and there is the free particle type exp(-iEnt). We suggest that the system is characterized by uncertainty in space and time with En being a known constant at each x. This suggests that various free particle p (momentum) values combine to create the wavefunction such that -1/2m d/dx d/dx W/W is average kinetic energy which in turn adds to V(x) to create a constant En. We are thus concerned with uncertainty in time and space and not in momentum. In the limit of high energy values, a crest fits within a tiny dx and prms is essentially the classical p value which is considered certain, even though in quantum mechanics the range of a(p) weights hold for all n (W(x)=Sum over p a(p)exp(ipx)). Thus we treat a high n limit as being completely certain and not having entropy. This differs from the approach of using C/v(x), where v(x) is velocity given by .5mv(x)v(x)+V(x)=En, as a probability and calculating Shannon’s entropy using this.
Interference between exp(ipx) momentum eigenstates in W(x)=wavefunction= Sum over p a(p)exp(ipx) or bound eigenstates in W(x,t)=Sum an(En)exp(-iEnt) Wn(x) are well known in quantum mechanics. The resulting interference (seen in W*(x,t)W(x,t)) seems to be the effect of an OR probability condition i.e. adding complex probabilities with both the real and imaginary terms being periodic and containing both positive and negative values. (Here the wavefunction is treated as a kind of probability (i.e. a conditional probability.) In this note, we wish to argue that interference does not always occur. In particular, a free particle has a wavefunction of exp(-iE(p)t) exp(ipx where E=pp/2m, but only exp(ipx) terms interfere in W(x)=Sum a(p)exp(ipx). exp(-iE(p)t) terms do not interfere, rather a new time cycling resonance is created described by an overall exp(-iEt) where E represents overall energy (kinetic and potential). In other words it seems that a new kind of energy is created namely potential energy which must be considered together with kinetic energy. A particle in a bound state, treated as a free particle for statistical purposes, maintains a kind of identity and energy flow independent of the potential. Its energy identity, however, described by exp(-iE(p)t) where E(p)=pp/2m is transformed as it now interacts with exp(ikx) terms of V(x) to contribute to an overall new cycling resonance of exp(-iEt). Thus, exp(iE(p)t) terms do not interfere. For a linear combination of bound states (e.g. in a kind of heat reservoir) W(x,t)= Sum over n an(En)exp(-iEnt) Wn(x) and a new cycling resonance is not created, rather the reservoir kicks one n state to another and exp(-iEnt) terms interfere. Interference gives rise to the humps or troughs found in the original exp(ipx) or exp(-iEnt), but no longer yields a modulus of 1 at each x or t point. Thus interference shows the nonlocality in the problem. exp(ipx) on the other hand contains both locality and nonlocality. There is a wavelength proportional to 1/p and cos(px) and sin(px) are both nonlocal, but the modulus of exp(ipx) is 1 which is a local property. Quantum mechanics allows both for interference which yields nonlocality and noninterference, e.g. exp(-iEnt) which contains both locality and nonlocality. We investigate this idea in this note.