Recently, there has been much activity related to statistical models which aim at describing quantum mechanics. It appears, however, that these models do not discuss the time periodic exp(iEt) nature of wavefunctions, at least in an explicit manner. The purpose of this note is to attempt to examine this periodic nature even though it explicitly disappears when calculating the time independent density. To do this , high and low potential V(x) limits are considered. Given that energies of an oscillator are multiples of a physical frequency (plus a background term) and that sound waves can be absorbed and emitted by artificial atoms (quibits), it seems that the physical nature of exp(iEt) may be important.
Recently, there has been much activity related to statistical models which aim at describing quantum mechanics. It appears, however, that these models do not discuss the time periodic exp(iEt) nature of wavefunctions, at least in an explicit manner. The purpose of this note is to attempt to examine this periodic nature even though it explicitly disappears when calculating the time independent density. To do this , high and low potential V(x) limits are considered. Given that energies of an oscillator are multiples of a physical frequency (plus a background term) and that sound waves can be absorbed and emitted by artificial atoms (quibits), it seems that the physical nature of exp(iEt) may be important.
We consider the possibility that both classical statistical mechanical systems as well as quantum mechanical time independent systems evolve according to the Hamiltonian H. This suggests that given the energy of the system, the Hamiltonian operating on a density or wavefunction will bring back the density or wavefunction with a weight factor of E, energy. For the quantum mechanical case, E represents a frequency. We argue that it is a frequency for both the quantum and statistical mechanical case, related to how quickly the system interacts under the Hamiltonian. We also examine an aspect of interference in the quantum wavefunction which may be related to a 2x2 model of a particle which cycles back and forth as it moves with an average velocity v in a certain direction. We argue that the plane waves stirred up in a potential (or in the Fourier series of the wavefunction) are associated with periodic time motion (due to zitterbewegung). We argue that as these plane waves interfere, they create a new constant average energy throughout a system with a potential which acts as the period of periodic time behaviour, namely exp(iEt)
The classical limit of quantum mechanics is often considered in terms of the time dependent Schrodinger equation. In this note we consider the bound case solution i.e. a time dependence of exp(-iEt) and as in previous notes argue that time has been removed from this statistical equilibrium picture as a the quantum particle tries to establish a kind of impulse potential field based on superpositions of exp(ipx) terms. Such a potential field is linked with a bound state and does not involve following the particle about in time as it undergoes stochastic impulse hits with exp(ikx) terms from V(x)=Sum over k Vk exp(ikx). On average a classical energy balance equation KE(x)+V(x) = En holds for every energy En, but the average is created by a spread of momenta i.e. W(x)=wavefunction= Sum over p a(p)exp(ipx). In this note we consider the case of a particle in an infinite potential well and note how the spatial resolution becomes very high at the same time that the momentum spread becomes very narrow in the case of high n (energy level). Thus as n becomes approaches infinite, it is not just the vrms(x) which is linked with x, but the statistical scenario of quantum mechanics approaches a situation in which x is being resolved to a very high level and p is almost constant. Furthermore quantum entropy becomes a constant (independent of n) as n→ infinite suggesting a theory which is nonstatistical in this limit.This would represent a classical picture except for the fact that time is still removed from the quantum bound state i.e. forward and backward exp(ipx), exp(-ipx) still interfere. Given that x and p are well resolved locally for high n it seems one may follow the high n quantum particle in space and time and manually remove the forward and backward portions to create a theory of a particle moving in one direction under V(x) i.e. classical physics. This bypasses the idea of a wavepacket whose x-spread increases in time and allows one to base classical physics on a bound state solution of a quantum particle in an infinite well.
In classical mechanics, the spatial density d(x) of a particle moving under a conservation of energy law .5mv(x)v(x) + V(x) = E is proportional to 1/v(x). This is obtained from d(x)= dt/T, i.e. the probability to be at x equals the time spent at x divided by the cycle time T. Given that dx=vdt to first order, density is proportional to 1/v(x). Now, d(x) is supposed to match the time independent quantum mechanical density W(x)W(x), where W(x) is the wavefunction, for high energy levels. Thus, one would expect the time a p wave (plane wave) spends at a point should be part of time independent quantum mechanics, but it does not seem to be overtly present. Furthermore, there is the question of T for the quantum mechanical problem. It is known that even for a time-independent solution W(x) has a factor exp(iEt). Thus, it seems time is present, even if it disappears in the density W*(x)W(x). It is argued that W*(x)W(x) = P(x) where P is probability, but W(x) is related to conditional probabilities such as P(x/p) or P(p/x). Thus, these conditional probabilities should be related to time. In an earlier note, we tried to show the time-independent Schrodinger equation implies all p plane waves are cycling with the same frequency E. In this note, we try to find a quantum mechanical representation for P(x,p)=dt/T where dt(x,p) is the time spent at x by a plane wave p and also try to evaluate T.
In the literature, stochastic approaches employed to derive the Schrodinger equation seem to focus on the idea of a classical spatial density which depends on time and space ie. density(x,t). As a result, d/dt (partial) density appears in various equations such as the continuity and Focker Planck equations. At one point in the derivation the quantum wavefunction is introduced “mathematically” as W(x,t)=sqrt(density(x,t)) exp(i S(x,t)) where S is related to a velocity field through u(x,t)=velocity= d/dx S. From quantum mechanics, -Et becomes a term of S the velocity field. The stochastic/hydrodynamical approaches employ the term du/dt partial = d/dt (partial) d/dx S and then “pull out”” d/dx leaving d/dt (partial) S which yields -E average energy from the -Et term. In this note we argue that for a bound quantum state density(x) so d/dt (partial) density = 0, yet the statistical ideas of quantum mechanics should be present in such a case. Furthermore for the free particle wavefunction exp(ipx-iEt), density=1. In fact, the Schrodinger equation should follow for a free particle even the case of density=1. Stochastic processes seem to be based on equations which consider a change in classical spatial density at a point x in time due to flow in or out or stochastic changes. Quantum processes seem to deal with periodic probability changes from one dimension to another exp(ipx) (2 dimensions) which are somehow related to a physical process of momentum flow. Ultimately one links this process to an energy conservation which does not seem to be the same as a classical stochastic/hydrodynamical approach. In this note we argue that a free particle with a constant momentum p has periodic stochastic processes associated with it through its complex probability change in the region of its wavelength, but that these are linked to creating the momentum p. Furthermore momentum is linked to energy so we argue that the Schrodinger equation is a stochastic equation linking energy and momentum and that time and space are separated i.e. there is no situation of following a change in spatial density in time for a free particle with momentum p. For a bound state, one has a distribution of free particles, but again time is removed from the problem in the sense that it only appears in exp(-iEt) in the wavefunction to again establish a relationship between energy and average momentum and potential. There is again no nonzero d/dt (partial) density term which seems to be key in stochastic equations. Only if one takes a linear combination of energy eigenstates Sum over En an(En) exp(-iEnt) Wn(x) does one obtain the form density(x,t), but we argue that quantum features should already appear in exp(ipx) and the bound state scenario with a zero d/dt (partial) density(x,t). We argue, however, that the Schrodinger equation and fundamental stochastic ideas of quantum mechanics should already be present at the level of exp(ipx).
In a previous note (1), it was shown that for a ground state oscillator wavefunction, one could analyse the problem in momentum space as a free gas of momentum particles and obtain exp(-p*p/2mT) as fp*fp, fp being the momentum wavefunction. A free energy equal to zero was obtained, with kinetic energy being the energy term in E-TS. This suggests quantum particles scatter (with conservation of momentum) like classical point particles in momentum space, but differ from classical point particles in position space. In particular, one takes a Fourier transform of fp as it seems that each momentum wave carries an exp(ipx) function which is a sort of “baggage” term, possibly due to zitterbewegung. This seems to explain why a free gas in momentum space for the ground state oscillator yields a Gaussian density with W(x)=exp(-ax*x) (W(x)=wavefunction). An oscillator potential is then needed to keep this density with its momentum space pressure in balance. Thus, quantum mechanics and classical statistical mechanics become very similar for this scenario. This begs the question: What about other potentials or oscillator states higher than the ground state? In such a case, it still seems that one may consider “particles colliding in momentum space” with each carrying an exp(ipx) “baggage” term, but one does not have a “classical statistical mechanical” equilibrium scenario any longer. Rather, it seems that one has a net force created by -dV/dx (V(x)=potential) and a difference in pressure d/dx [Average(p*p)d(x)] where d(x) is density. There also seems to be an average velocity (1/W)dW/dx. In momentum space such a velocity would not exist for f(p)=f(-p), but because of the baggage of exp(ipx) it is present i.e. p exp(ipx) - p exp(-ipx). Newton’s law, with force acting on this velocity appears in the equation d/dx [Average(p*p)d(x)]=F(x) d(x). This same equation is equivalent to the gradient of the time-independent Schrodinger equation. For quantum mechanics, one has one particle rather than many as in classical statistical mechanics. Thus, it seems that this “Newton’s law” balance is based on a long term time average. It seems, however, that there should be a period T during which the average can establish itself. After this, one would have periodic time motion where the system keeps scattering back into itself in this characteristic time. This period can be deduced, it seems, from the time-independent Schrodinger equation and is found to be E, the energy.
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.