In the literature (1) it has been noted that (d/dx W)/W - (d/dx W*)/W* represents a classical velocity, where W is the wavefunction. For bound states, this expression is zero, yet (d/dxW)/W seems to play the role of density flux as d/dx (WW) = 2W d/dxW. When multiplied by E, the energy, it becomes and energy flux as there is constant energy at each point x. In this note, we wish to consider some examples related to this flux. In particular, for the case of V(x)=0 (no potential), the flux becomes a solution of the time independent Schrodinger equation. This and other considerations may suggest that periodic type functions (possibly related to photons or phonons) when multiplied by the wavefunction W, can lead to a new, higher energy wavefunction. We also consider the role of energy flux as it relates to discrete energy levels in quantum mechanics.
In the literature (1) it has been noted that (d/dx W)/W - (d/dx W*)/W* represents a classical velocity, where W is the wavefunction. For bound states, this expression is zero, yet (d/dxW)/W seems to play the role of density flux as d/dx (WW) = 2W d/dxW. When multiplied by E, the energy, it becomes and energy flux as there is constant energy at each point x. In this note, we wish to consider some examples related to this flux. In particular, for the case of V(x)=0 (no potential), the flux becomes a solution of the time independent Schrodinger equation. This and other considerations may suggest that periodic type functions (possibly related to photons or phonons) when multiplied by the wavefunction W, can lead to a new, higher energy wavefunction. We also consider the role of energy flux as it relates to discrete energy levels in quantum mechanics.
Early experiments linked to bound state quantum mechanics noticed discrete energy E levels. (Ignoring the issue of radiation from accelerating charges), classical physics predicts continuous E levels. If one tries a statistical approach i.e. thinks in terms of a “perceived” density dt=C dx/v(x) where v(x) is velocity, one again has continuous energy levels. Thus, experimental results may suggest a non-classical condition which affects both acceptable densities and energy levels. Bypassing Newtonian mechanics which traces a particle through x(t), we note that even classically a potential interacts with p and not v i.e. pp/2m - popo/2m = -V(x)+V(xo). This equation holds for p= mo v = (.5mo) 2v etc. Thus, energy flow i.e. p is of interest and if one does not trace the particle as x(t) or consider acceleration (which is not considered statistically), one may view energy flux p as being “spread” over the x axis, it seems. The length 1/p then corresponds to one unit of energy flux for the case of p. In other words, each p is associated with a length 1/p. If this length may be physically resolved, one immediate trial result suggests a spatial probability which varies over 1/p. In other words one may have a periodic function such as cos(px). This physical scenario seems to arise only when the quantum particle interacts with a potential. Calling 2*3.14/p = wavelength, this length is not associated with the velocity of the particle, but with energy flow. Each particle has an identical type of cycling within its own wavelength i.e. cos(px) behaviour as each wavelength represents one unit of energy flux. The wavelength limits resolution and leads to a probability scheme. In a bound quantum system dx associated with V(x) is usually much smaller than the larger wavelengths and so classical ideas such as p and pp/2m still apply, but are modified i.e. multiplied by probabilities which may be positive or negative. Having a spatial probability linked to density which also creates a probability scheme used to calculate average kinetic energy at each x based on 1/p type wavelengths is independent of classical physics. Trying to force average kinetic energy at x in the wavelength scheme to match KE(x) classical leads to discrete E values.
In a previous note (1), we argued that the Maxwell-Boltzmann (MB) distribution exp(-p*p/2mT) and similar forms such as exp(-e1/T) and exp(-V(x)/T) were based on reactions of the form P(p1)P(p3)=P(p2)P(p4) or P(p1)V(1 to 2) = P(p2)V(2 to 1) ((1)) where p1, momentum, is being scattered into p2. Taking the natural log of this expression and comparing it to a conservation of kinetic energy (or energy levels in other cases) leads to the form exp(-p*p/2mT). This approach was applied to static statistical mechanics with a potential V(x), statistical mechanics with time independent flow, quantum mechanics with a temperature and fixed energy levels and the T=0 ground state quantum oscillator. It was argued that the MB type distribution breaks down for other bound state cases because there are two velocities in such cases. The first is the vrms(x) velocity where .5mvrms(x)*vrms(x) is the classical kinetic energy and u(x)=(1/W(x)) d/dx W(x) where W(x) is the bound state wavefunction. In this note, we try to give a specific calculation involving velocity (or flux) for the statistical mechanical case to show how it complements a reaction balance ((1)) as long as the flux velocity obeys Newtonian mechanics. For the ground state quantum oscillator, this is the case for the flux 1/W d/dx W, but not for other levels or potentials. Thus, this lack of Newton’s law in the flux 1/W d/dx W (which is responsible for d/dx d(x), where d(x) is the spatial density) seems to lead to the breakdown of the reaction balance ((1)) and the MB distribution in quantum mechanical bound states, leading to a new kind of statistical requirement for quantum bound states.
In bound state quantum mechanics, one has statistical observables (based on P(x) where P is probability) such as spatial density d(x), potential energy density V(x)d(x) and kinetic energy density .5mv(x)v(x) d(x), where v(x) is the classical velocity obtained from .5mv(x)v(x) + V(x) = E. (Kinetic energy density can also be written as -1/2m W(x) d/dx d/dx W(x) where W(x) is the wavefunction). The energy of a state can be obtained by integrating these statistical observables i.e. Integral dx d(x) V(x) + Integral dx d(x) .5mv(x)v(x) = E. If one performs this integration for two different states, one may calculate the energy difference. This begs the question: why does quantum mechanics deal with conditional probabilities such as P(p/x) directly related to W(x)? The issue perhaps begins with the appearance of V(x) and E in the time independent Schrodinger equation. Kinetic energy (not kinetic energy density) should appear and be somehow linked to some motion. A question arises as to why the energy density is proportional to the spatial density i.e. Energy density = E d(x)? This is directly linked to the time-independent Schrodinger equation, but one may try to argue that classical energy conservation .5mv(x)v(x) + V(x) = E at each point x should hold in quantum mechanics with .5mv(x)v(x) equal to Sum over p p*p/2m P(p/x) where P(p/x) = fp exp(ipx)/W(x). If one, however, considers excited states written as Wn(x)= Wo(x)Hn(x) where Wo(x) is the ground state wavefunction, one finds that: -1/2m d/dx d/dx Hn(x) - 1/m [d/dx Wo(x)]/Wo(x) [d/dx Hn(x)] = E1 Hn(x) ((1)) In other words, an eigenvalue equation appears again with no V(x) present. If one assumes Hn(x) has physical meaning (i.e. is related to a phonon or photon), then ((1)) may suggest there is a physical principle in place, i.e. the idea that one must have a “constant” energy at each point x. Here E1 is strictly kinetic energy (although not all of it). This “principle” has consequences as it restricts the values of E1 (i.e. discrete eigenvalues) and leads often to a periodic Hn(x) which has direct effect on kinetic energy density and potential energy density. Thus, having Hn(x) interact with Wo(x) to produce a constant energy E1 times Hn(x) does affect kinetic and energy densities at each x point, just not kinetic energy itself (it is constant i.e. E1). The purpose of this note is to examine this possible principle behind quantum mechanics.
In the literature (e.g. (1)) the Schrodinger equation is sometimes derived using classical fluid equations which follow from Boltzmann’s transport equations. In particular a continuity equation and a conservation of momentum equation involving the potential dV(x)/dx are used. We have argued in Part I that the continuity equation leads to 0+0=0 for a free particle wavefunction exp(-iEt+ipx) because density is 1. Furthermore a bound quantum state also leads to 0+0=0 partly because d/dt partial density(x) =0. Following the analysis of (1) for which a new field S(x,t) is introduced through u(x,t) =velocity field = d/dx partial S(x,t) one may see that the solution S(x,t)= f(t) for any function f yields u(x,t)=0 for a bound state. In fact, for a bound state, the continuity equation contributes nothing to the derivation of the Schrodinger equation except for the result density(x) (no t present) which is known already. The Schrodinger equation for the bound state follows completely from the fluid transport equation containing dV/dx. For this to occur, however, one must use S(x,t)=-Et. It seems, however, that for a bound state with u(x,t) the natural approach is to have S(x,t) =0 Setting S(x,t)= -Et is equivalent to having exp(-iEt) appear in the wavefunction and we argue in this note that this is an independent assumption which does not follow from fluid mechanics. In particular, we link -Et for the bound state to A(x,t)=classical action = -Et+px. For a bound state there is an average energy value, but overall p average is 0. For the free particle quantum system we argued that two flow flux equations d/dx partial A = p ((1a)) and d/dt partial A = -E ((1b)) govern the system. These may be written separately as eigenvalue equations yielding: id/dt partial exp(iA) = E exp(iA) ((2a)) and -id/dx partial exp(iA) = p exp(iA) ((2b)). We show that exp(-iEt) for the bound state is a built-in assumption needed to make the fluid mechanical equation approach to the Schrodinger equation work. As a result, this assumption id/dt exp(iA)=E exp(iA) needs to be presented together with the two fluid equations, otherwise one does not derive the Schrodinger equation. In fact, one may use ((1a)) and ((1b)), the two flux/flow equations we propose for a free particle and create a momentum distribution P(x/p) = a(p) exp(ipx)/W(x) where W(x)=wavefunction = Sum over p a(p) exp(ipx). Using conservation of energy at each x point: [Sum over p a(p)pp/2m exp(ipx)] / W + V(x) = E leads to the Schrodinger equation without the use of fluid equations.
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), we argued that both free particle quantum and classical mechanics follow from an energy flow/flux scheme based on: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) A(x,t) turns out to be the classical action tL(v) (L=Lagrangian) with x/t=v and the above hold for both the relativistic and nonrelativistic cases. We further argued that ((1a)) and ((1b)) lead to two sets of solutions, one classical and the other quantum. The classical solution assumes dt→0 and dx→0 i.e. flows may be followed at each point in space (and time). In (2) we argued that ((1a)) and ((1b)) are consistent with p=dL/dv (partial) and H=E= pv - L. From these together with p=Ev one may find: E=mo/sqrt(1-vv) (c=1). For the nonrelativistic case, E= mocc + .5mvv. In this classical analysis one may resolve each t and x, thus if one has a function V(x) (potential) representing energy, the nonrelativistic result is: (E-V(x)) = .5mvv. The quantum solution to ((1a)) and ((1b)) is very different. It establishes eigenvalue equations i.e. id/dt (partial) exp(iA) = E exp(iA) and -id/dx (partial) exp(iA) = p exp(iA) for the free particle. The importance of this is that new physical mechanisms seem to come into play. A= -Et+px so exp(-iEt+ipx) suggests time and spatial regions proportional to 1/E and 1/p in which there is some ambiguity i.e. there are two dimensions and there seems to be some kind of probability for the particle to be in one dimension or the other at x (and similar results for t) i.e. cos(px) and sin(px). As a result, one may not have dx→0 and dt→0. Furthermore, it is argued that a two dimensional associated probability scheme must accompany energy flow. At first this may seem strange, but if one considers a photon which has an energy flow Ev, one has electric and magnetic fields perpendicular to each other and perpendicular to the motion which undergo periodic motion linked to cos(wt) and sin(wt). Thus, the quantum particle periodic probability mimics an already known result from Maxwell’s equations. In fact, we have argued that for fermions there is additional angular motion i.e. spin linked with the recognition of 3-space. Thus we argue that quantum mechanics is involved with periodic type probability motions which are needed to resolve dx and dt (linked to exp(-iEt+ipx) and periodic probability motion (spin) and are also needed for fermions to resolve 3-space as two p vectors may have point in different directions (i.e. spin). Quantum mechanics therefore seems to be a very different theory from classical mechanics, but both appear to be based on energy flow/flux. As a result, the average of a quantum bound state should be linked to a classical system we argue in this note as one only deals with energy flow/flux.