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 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(-pp/2mT) as fpfp, 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(-axx) (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(pp)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.
Both quantum bound state equilibrium and classical statistical mechanical equilibrium are characterized by d/dt partial density(x,t)=0. For a quantum particle, W(x,t)=exp(-iEt)W(x) and d(x,t)=W*W. Density is linked to fluid flow and arguments of pressure balance such that the overall average fluid flow is zero are made. This pressure-force balance is common in classical statistical mechanics and used to calculate the variation of density with x i.e. obtain a differential equation with d d(x,t) /dt partial. A similar balance equation is used in (1) as the authors try to derive the Schrodinger equation using a fluid dynamical approach. In this note, we try to argue that one may think of the two equilibria in terms of an energy quantity which remains constant throughout space. This energy quantity depends directly on the type of interactions in the system at any point x. For a classical ideal gas, balanced two body time reversal scattering represents the interaction. V(x), a potential, does not change this fundamental manner of interacting at any given point, although it does affect density. As a result the energy related constant at each x is T, related to average kinetic energy. We argue that this already points to the dominant type of interaction creating the statistical interaction i.e. one that depends only on p in interactions which conserve kinetic energy. This “happens” to be equivalent to maximizing Shannon’s entropy, but we argue this is particular to this scenario in the sense that not all interacting systems will have this equivalence. In other words, we argue that one must be careful with regards to the maximization of entropy as a fundamental rule. In particular, we consider the single particle bound quantum state. The interaction exists between the particle and the potential V(x), but one may consider a distribution of free particle momentum probabilities i.e W(x)=wavefunction = Sum over p a(p)exp(ipx). The form exp(ipx) comes from separate arguments as is discussed in the note, but the point is that there are no two body scattering reactions occurring at a given x, neither is there acceleration from on x to another (except for a vrms mathematical quantity). Nevertheless the “free particle” exp(ipx) interacts at each x with V(x). As a result of this interaction, one has free particles exp(ipx) which are no longer associated with pp/2m, but rather with an Eave which is a constant for every p. In other words the identity of exp(ipx) is unusual because it is linked to exp(ipx) and pp/2m, but also to Vk1 a(p-k1), Vk2 a(p-k2) etc where V(x)=Sum over k Vk exp(ikx). Thus each x point is associated with a constant Eave and this defines the equilibrium. (It is also equivalent to conservation of energy at each x.) In such a case, one does not maximize entropy to find a(p) values. In the two cases, the common point seems to be considering the interactions at each x and finding a constant “energy” type quantity which does not vary with x. This we argue appears to define equilibrium for both the classical statistical and quantum bound states. Instead of considering density which does not change in time at each x, there is constant energy which not only does not change in time, but has a constant value at each x which seems to be an alternative rule.
Newtonian force is based on a change in momentum in time, but momentum is a vector and so may change in magnitude and direction. In many cases, the change is of one form or the other. For example, for a bound particle the change is in kinetic energy i.e. in magnitude, while for a particle moving at constant speed in a circle or pressure in a Maxwell-Boltzmann gas, the change is in direction. For an MB gas, the probability for p and -p are the same so the average momentum at any point x is 0 and thus chunks of the gas do not move, even though a single particle in the gas moves. The pressure, on the other hand, is the average of 2pv (2p for impulse, v for flux) and so pressure for p and -p do not cancel. A quantum free particle is based on exp(ipx) which does not contain time and is a dynamical probability, This is both an eigenfunction of the momentum operator -id/dx and the kinetic energy operator -1/2m d/dx d/dx, yet as pointed out in Part II, there are two pieces (cos(px),sin(px)) with a spatial density and hence a kind of pressure/force. These, however, combine to give the sense of constant motion in one direction. We argue this is due to a dynamical probability, unlike a static balance of pressures/forces found in an MB gas or Newtonian statics. There is a density with pressure distribution cos(px), but this is continuously changing because the particle moves in one direction with constant p. Thus the two pressure distributions “balance” so to speak to create a constant p. Interference of exp(ipx) and say exp(-ipx) may “bring out” a “static” spatial density i.e. 2cos(px), but behind the scene the dynamical probabilities still exist. It is these which “hold” the cos(px) together which appears strange from a static point of view. For example, a quantum bound state has a static W*(x)W(x) spatial density at the same time that the single particle has an average classical kinetic energy KE(x)=En-V(x). Interference also gives rise to the notion of an impulse-pressure in quantum as one ignores the isin(px) and i sin(-px) terms. Creating a pressure scenario with p and -p leads to two considerations. First, there is an impulse-pressure (defined without flux) at each x point and secondly, this impulse-pressure changes with x. The time-independent Schrodinger equation is formulated in terms of energy (i.e. a scalar): KE(x)+V(x)=En where KE(x)= {-1/2m d/dx dW/dx} \ W and W=Sum over p a(p)exp(ipx). Thus, as in Newtonian mechanics, one thinks of -dV/dx solely accelerating a particle. The quantum particle however, is described by an ensemble W(x) which has pressure. This pressure is due to a combination of the form exp(ipx) and V(x), so V(x) does not simply change root mean square speed. exp(ipx), by itself, creates pressure when interference occurs as in a two slit scenario. We try to examine these ideas in this note.
In a series of notes (1), we argued that the time-independent Schrodinger equation may be considered as a statistical conservation of average energy equation. We further argued that V(x)=Sum over k Vk exp(ikx) delivers stochastic potential hits which create a momentum distribution W(x)= Sum over p a(p) exp(ipx) at each x. The form exp(ipx) describes a free particle and a statistical distribution is made of such free particles. The form exp(ipx) has a modulus of 1 indicating that P(x)=constant. d/dx exp(ipx) = ip suggesting the change in conditional probability with space is related to classical momentum. Given a momentum distribution, each p at x occurs at a different time. Thus, one does not follow a classical particle in time x(t), nevertheless an average energy conservation equation applies. Taking the time-independent Schrodinger equation E W = -1/2m d/dx d/dx W + V(x) W and writing EW = i d/dt exp(-iEt), one obtains a time-dependent equation. We argue that if the time-independent case represents a classical conservation of average energy, the time-dependent equation should be linked to Newton’s equation of motion. This immediately leads to a problem because we argue one does not want x(t) for a statistical system. As a result, x in the time-dependent Schrodinger equation, which represented spatial position before, now represents a spatial variable that is independent of time. We consider the oscillator case and note that an amplitude is such a candidate where x(t)=XA(t). Thus, the time-dependent Schrodinger equation ultimately becomes an equation in A(t) with the amplitude cancelling, at least in the free particle and oscillator case. From such a consideration, one may work “backwards” and find a function for which the time-dependent Schrodinger equation (equating powers of X) becomes Newton’s second law. From this, one may show that exp(i Action) where Action = Integral dt1 (0,t) L where L is the classical Lagrangian is this solution with d/dX applying to X which is an amplitude, not x(t).
The classical Lagrangian L leads to Newton’s second law which is equivalent to an energy-momentum conservation equation. The classical action: A = Integral dt1 L(t1) for a constant velocity v=X/T may be varied with respect X and T i.e dA/dT = -E and dA/dX= p. Thus, one may create a differential equation in (dA/dT)(dA/dt) and dA/dx dA/dx for the EE=pp+momo equation. Alternatively, one may create other differential equations. Quantum mechanical differential equations, however, are quite different with exp(i Action) being a solution. Even this may be written as the solution of a non-Schrodinger equation i.e. id/dt exp(iAction) = -1/2m (d/dx exp(iAction)) (d/dx exp(iAction)) for a free particle. For a constant velocity, exp(iAction) is equivalent to exp(ipx-iEt). We argue that this solution is statistical because it separates x and t (as independent variables) and holds for all x and t. Thus, quantum mechanics seems to be a statistical approach using free solutions exp(ipx-iEt) to create an ensemble as in classical statistical mechanics. A difference, however, is that exp(ipx-iEt) “interferes” even if one only takes the real part cos(px-Et). We consider the case of a particle in a box with infinite potential walls to argue that the periodicity is warranted i.e. positive and negative values (interference). Even though the modulus is 1, exp(ipx-iEt) may be operated on by various generators e.g. d/dx the generator of translations. This is analogous to operations on vectors in linear algebra. As noted above, quantum equations are not only energy-momentum conservation equations. They yield a wavefunction W(x) which is an unusual momentum distribution W(x)=Sum over p a(p) exp(ipx). exp(-iEt) holds overall because average E is the same at each x. Thus, one needs a W(x) which leads to a momentum-energy conservation equation, but also satisfies certain boundary conditions which become linked to spatial density= W*W. This leads to discrete energy levels and also cases of humps and troughs in W(x). We attempt to investigate some of these ideas in this note.
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 Part I, we considered the ideal gas law P= density(x) RT of a Maxwell-Botlzmann gas, which follows from the notion of impulse 2p multiplied by v (flux) and then averaged,and examined how it balanced (on two sides of a box) a force -dV(x)/dx multiplied by density. We argued that in a quantum bound state there is no notion of internal time and hence no flux and so considered an average impulse 2p multiplied by an overall internal time in the system, namely hbar/E. We found that this scenario balanced with -dVd/x matches the quantum oscillator ground state. In this note we ask: What about other potential V(x) situations? Quantum mechanics seems to be a mixture of statistical mechanics and Newtonian dynamics i.e. the time-independent Schrodinger equation is, on average, an energy conservation equation for all energy levels i.e. KE(x)+V(x) = En where KE = {Sum over p a(p)pp/2m exp(kpx)}/{Sum over p a(p)exp(ipx)}. In (1) we took d/dx to bring out a force term -dV/dx and showed if is linked to <ppp/2m> and <pp/2m> , but analyzed these in terms of fluxes. The notion of flux makes sense using external time, but internally there is no time and so here we try to examine the situation in terms of various impulses.