Traditionally, a quantum bound state is said to approach classical behaviour for large n (from En=energy). The problem is that peak and valley spatial density regions still exist although an envelope passing roughly through the peaks of Wn(x)Wn(x) (where Wn is the wavefunction) matches C/v(x) where v(x) is the classical spatial density. This is proportional to delta t the time a particle spends within constant sized delta x pieces of a bound path. Alternatively, a wave packet is frequently used, but its width tends to increase in time. In a previous note (1), we argued that one may write the free particle classical action (relativistic or nonrelativistic) as A= -Et+px with t and x being independent. This leads to dA/dx partial = p and dA/dt partial = -E. Creating eigenvalue equations -i d/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = Eexp(iA), one may see that exp(-iEt) is periodic in two dimensions in time and exp(ipx) in space. In this note we examine whether the separation of x and t suggests two schemes for the spatial region. The first scheme is linked to exp(-iEt) where E is average energy at each x. In such a case E=pp/2m (for a free particle) at each x and x=p/m t. exp(ipx) on the other hand demonstrates fluctuations in two dimensions within this region which would not normally be seen in the classical realm due to lack of resolution. If one considers a bound system, these two schemes become clearer. exp(-En t) is associated with an energy En equal to KEclassical(x) + V(x). On the other hand there is the quantum momentum distribution Wn(x)=Sum over p a(p)exp(ipx) such that: [Sum over p pp/2m a(p)exp(ipx)]/Wn(x) + V(x) = En ((A)) As a result, the momentum distribution Wn(x) is linked to En at each x through ((A)). The two dimensional waves exp(ipx) interfere to create an rms v which is classical. This classical motion exists for all n. Wn(x)Wn(x) on the other hand demonstrates quantum fluctuations, but these include also an rms probability of C/v(x) where v(x) is the classical velocity. For large n, C/v(x) becomes the envelope of Wn(x)Wn(x) and so quantum fluctuations are within the envelope. For lower n we argue that C/v(x) must be “removed” from Wn(x)Wn(x) to see the full quantum fluctuations, but that C/v(x) still applies to the classical rms motion which is present and associated with exp(-iEnt). In other words we use neither wavepackets or the idea that classical mechanics is involved only for large n. Classical mechanics is present for all n in ((A)).
Traditionally, a quantum bound state is said to approach classical behaviour for large n (from En=energy). The problem is that peak and valley spatial density regions still exist although an envelope passing roughly through the peaks of Wn(x)Wn(x) (where Wn is the wavefunction) matches C/v(x) where v(x) is the classical spatial density. This is proportional to delta t the time a particle spends within constant sized delta x pieces of a bound path. Alternatively, a wave packet is frequently used, but its width tends to increase in time. In a previous note (1), we argued that one may write the free particle classical action (relativistic or nonrelativistic) as A= -Et+px with t and x being independent. This leads to dA/dx partial = p and dA/dt partial = -E. Creating eigenvalue equations -i d/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = Eexp(iA), one may see that exp(-iEt) is periodic in two dimensions in time and exp(ipx) in space. In this note we examine whether the separation of x and t suggests two schemes for the spatial region. The first scheme is linked to exp(-iEt) where E is average energy at each x. In such a case E=pp/2m (for a free particle) at each x and x=p/m t. exp(ipx) on the other hand demonstrates fluctuations in two dimensions within this region which would not normally be seen in the classical realm due to lack of resolution. If one considers a bound system, these two schemes become clearer. exp(-En t) is associated with an energy En equal to KEclassical(x) + V(x). On the other hand there is the quantum momentum distribution Wn(x)=Sum over p a(p)exp(ipx) such that: [Sum over p pp/2m a(p)exp(ipx)]/Wn(x) + V(x) = En ((A)) As a result, the momentum distribution Wn(x) is linked to En at each x through ((A)). The two dimensional waves exp(ipx) interfere to create an rms v which is classical. This classical motion exists for all n. Wn(x)Wn(x) on the other hand demonstrates quantum fluctuations, but these include also an rms probability of C/v(x) where v(x) is the classical velocity. For large n, C/v(x) becomes the envelope of Wn(x)Wn(x) and so quantum fluctuations are within the envelope. For lower n we argue that C/v(x) must be “removed” from Wn(x)Wn(x) to see the full quantum fluctuations, but that C/v(x) still applies to the classical rms motion which is present and associated with exp(-iEnt). In other words we use neither wavepackets or the idea that classical mechanics is involved only for large n. Classical mechanics is present for all n in ((A)).
In a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by 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.
In previous notes (1) we argued that the free particle action A (either nonrelativistic or relativistic) may have v (velocity) written as x/t and x and t varied independently to yield: dA/dx partial = p and dA/t partial = -E with A=-Et+px, the Lorentz invariant classical Action = t Lagrangian. We further argued that one may write eigenvalue equations i.e. -idexp(iA)/dx partial = p exp(iA) and idexp(iA)/dt partial = Eexp(iA) and call exp(iA) a type of probability. We noted that if a potential is present one may use P(p/x)= a(p)exp(ipx)/W(x) [W(x)=wavefunction = Sum over p a(p)exp(ipx)] and establish a conservation of energy equation KE(x) + V(x) = E such that there exists the same constant E at each x. We described this in two ways. First E represents binding energy so at each x one has the same binding energy i.e. each x point maps to the same E as part of the mass of an overall moving system. Secondly, we argued that E was a parameter indicating that equilibrium existed at each x point, much like temperature T. In this note we wish to consider a different idea suggested in (2), namely that if one has a quantum state: W(x)= Sum over n a(n,t) Wn(x) this may be written as: <i |H| j> a*(i,t)a(j,t) = <W H W>.. If Hamilton’s equations for complex variables are applied to these then: i d/dt partial a*(i,t) = d/da(i,t) partial { <W H W> } ((A)) (2) argues this is the time-dependent Schrodinger equation. In other words, probability weights of the Schrodinger equations (which are complex) obey Hamilton’s classical equation. In this note, we point out that the formalism dA/dx partial = p already introduces operators into classical physics, so it follows that H is an operator which may act on states. We consider the idea of ((A)) as applied to a free quantum particle moving. Next, we apply ((A)) to a quantum bound state and argue that a classical Hamilton’s equation applied to d/dt a(p) partial may describe equilibrium. Finally ((A)) may be applied to a sum of eigenstates, but this is already shown in (2).
In a previous note (1) we argued that the free particle classical action A (relativistic or nonrelativistic) may be written in terms of v=x/t and then varied in x and t independently. One finds that dA/dx partial = p and dA/dt partial = -E implying A= -Et+px with x and t independent. This is the opposite of the classical mechanical idea of x(t). Creating an eigenvalue equation yields -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = Eexp(iA). exp(iA) plays the role of an unusual type of probability we argue which describes two dimensional periodicity (cos(px) and sin(px) ). We have argued that the quantum particle is the medium for a two dimensional wavelike motion of an energy-momentum wave. A wave type structure is needed to describe dynamics (like a classical wave), but unlike a classical wave two dimensions are required because there is no preference of any x point for a free particle. Both cos(px) and sin(px) both treat x in a preferential manner as their values vary with x. For a classical wave this is fine because the creation point of the wave imposes boundary conditions, but there is no such boundary condition imposed on a free particle. exp(ipx-iEt0 must somehow then link to P(x)=constant. One may note that exp(-iA)exp(iA) = 1 for the free particle. This may be argued to be linked to spatial invariance i.e. P(x)=constant where P(x) is not exp(ipx). Here we argue that exp(-iA)exp(iA) represents a product of two unusual probabilities , one for motion in a certain direction and the other for the time reversed scenario. If the quantum particle is the medium of an energy-momentum wave exp(-iEt+ipx), then the product of the regular and time reversed unusual probabilities cancel or undo each other. In other words one does not see any change in space as long as one has a noninteracting exp(-iEt+ipx) . Given that -id/dx is a linear operator this product yields a momentum of 0. We consider this idea and its implications on a momentum distribution Sum over p a(p)exp(ipx) which may represent a bound state or other states. In particular, it seems that P(x)=W*(x,t)W(x,t) which yields a classical type of probability (P(x) real, positive and normalized to 1) is linked to a product (merging) of forward and time reversed situations so that any spatial density which remain due to the quantum particle acting as a medium for an energy-momentum wave (as well as the time reversed part) would represent an actual change in spatial density. exp(ipx) already contains two periodic spatial functions (cos(px) and sin(px)), but does not manifest it unless there is an interaction.
In a previous note (1) we argued that both the relativistic A= -mo t sqrt(1-vv) and nonrelativistic A=t .5mvv classical free particle actions may be written with v=x/t and x and t varied independently to yield: dA/dx partial = p and dA/dt partial = -E. Thus A is equivalent to -Et + px for a free particle with x and t being treated as independent. We argue in this note that such an assumption is equivalent to a statistical picture for a free particle even at the classical level because x and t are treated as independent. Thus a constant p may occur at any x because no time is linked to x. Given that all space points are the same for constant p one expects a constant probability at all x points. Knowing p exactly means not knowing anything about which x one has. Secondly, one may create an ensemble of free particle p classical actions for a situation containing a potential V(x). Given that x and t are independent, there cannot be any notion of single p accelerating, although a vrms from an ensemble may. At a given x (x known exactly) one does not know which p is present. Thus the ideas of uncertainty of p and x follow immediately from this picture. If one considers a single particle and an ensemble of free particles in the x-t independent scenario, then each free particle action p is linked to A=-Et+px where E=pp/2m. This ensemble would arguably be stirred up by a potential V(x). If one wishes to consider an equilibrium scenario for which there is an overall En for the bound system one would expect an overall Ent for the system and for -pp/2m t pieces to disappear. Thus we postulate that pp/2m +{p part} G(p’s k’s) = En for all p where {p part}G(p’s, k’s) is an unknown function and V(x)=Sum over k Vk exp(ikx). Next one may consider the spatial situation for which V(x) is the classical potential for a single particle. The kinetic energy follows from an ensemble of free particles with the overall -Et removed. From classical considerations one would expect to have a KE(x). Thus one expects: KE(x) = {Sum over p a(p) f(px) } / {Sum over a(p)f(px) } with f(px), an orthonormal basis set, because the action contains px. (It is orthonormal because one wishes the identity of p to exist if not at a point, then over all of x.) We try to argue in this note that f(px)=exp(ipx), the momentum eigenstate which follows from A=-Et+px. We find G(p’s, k’s) which implies entanglement or mixing of a given p state with other p and k states from the potential V(x)=Sum over k Vk exp(ikx). Thus one expects an entropy linked to this momentum entanglement. Given the form exp(ipx), finding a set of ap’s which satisfy pp/2m + G(p’s, k’s)=En leads to a spatial distribution W(x)=Sum over p a(p)exp(ipx) so there should be an entropy contribution from this spatial part as well. In classical statistical mechanics one has both momentum and spatial contributions to entropy. We argue for S=Sp+Sx, but note that even though px has x, p symmetry as does the Heisenberg uncertainty principle var x var p >= hbar/2, Sp and Sx may not be symmetric in terms of important parameters such as n, energy level. In other words, Sp the entanglement entropy with V(x) may be the driver entropy sensitive to energy level changes while Sx the resulting spatial density may not be as sensitive. Furthermore the overall entropy Sp+Sx gives information about uncertainties in x and p as is already known in the literature where bound state Sp+Sx bounds are given.
We take as our starting point the free particle classical action which may be written as A= -Et+px if t and x are treated as independent variables. It is possible to find two relationships linking v (velocity) to x and t from dA=0. The first is the classical wave result vp = E while the second is p=Ev where E=mo/sqrt(1-vv) (c=1). Furthermore one may write dA/dx partial = p ((1a)) and dA/dt partial = -E. ((1b)) This may be generalized to two eigenvalue differential equations, but this immediately begs the question: Why do this? We argue that Lorentz invariant form -Et+px represents scaling of both t and x. For example, p scales x and we argue there should be physical relevance to this. This physical relevance results in the wavelength proportional to 1/p. In other words, -Et+px represents more than velocity. The treatment of the scaling for a quantum wave and classical wave differ, we argue, in that the former keeps E,t and p,x separate while a classical wave does not. In other words pv=E applies to a classical wave which is described by cos(-Et+px). In other words, one takes ((1a)) and ((1b)) which separate E,t and p,x into two independent equations and takes account of scaling by combining the two (to include both x and t) i.e. creating the classical wave equation: 1/vv d/dt d/dt partial W(x,t) = d/dx d/dx partial W(x,t). We argue that quantum mechanics on the other hand keeps ((1a)) and ((1b)) separate. The velocity (relativistic) p=Ev is not a wave velocity. A solution to ((1a)) is exp(ipx) and to ((1b)) exp(-iEt). These are usually combined for a free particle to give exp(-iEt+ipx) and a wave is described with a phase velocity pv(phase)=E. We prefer to keep exp(-iEt) and exp(ipx) separate . For example, for two slit interference one does not make use of phase velocity or exp(-iEt). For a bound state one uses p=Ev, but a single free particle is represented by exp(ipx) in W(x)=wavefunction = Sum over p a(p)exp(ipx). exp(-iEn t) appears after one has found En, using the exp(ipx)’s in the sense that one does not include an exp(- i pp/2m t) for each exp(ipx). We argue that by separating E,t and p,x, the latter leads to a scheme for which there is “no time”. It is true that the particle moves at a speed x/t, but p=m1v1=m2v2 etc so p does not uniquely define velocity, rather it defines an impulse hit 2p = Integral F dt . Thus the scaling px which leads to an x, p only wave exp(ipx) is linked to an interaction. If the particle is moving freely and not interacting, then one has classical determinism i.e. v=constant = x/t. As soon as an interaction occurs, one changes from variables x,t which are linked to x, P conditional (x/p) and there is uncertainty in the position of x within a wavelength. The phase in exp(ipx) is only dependent on x not on the time the particle reaches x. exp(ipx) thus represents a probability and one must consider a sum of different possible outcomes (as in a two slit experiment) in order to describe an interaction. This is very different from the local interaction of classical mechanics. Nevertheless for a bound state one makes use of the two formalisms i.e. KE(x)= Sum over p pp/2m a(p)exp(ipx) / W(x) = KEclassical(x). We try to discuss some of these ideas in this note.