Note: The equations d/dx partial [ T L] = p and d/dt partial [ TL] = -E hold for both the relativistic and nonrelativistic free particles because dL/dv = p and H=pv-L hold for both the relativistic and nonrelativistic free particle. In part I of this note, we argued that in quantum bound states one has a notion of dx and dt associated with p and E i.e. dx→1/p and dt→1/E, but not of continuous time associated with particle motion. exp(-iE t) contains time t, but this refers to a cycling time associated with various exp(ip1x) free particle probabilities combining with exp(ikx) to form exp(ipx). In other words there is a certain time resolution as well as a space resolution (wavelength) for the single particle and time t refers to the entire quantum resonance. In a bound system, there is also a vrms(x) which holds for every quantum energy state En and this is linked to a continuous space time picture i.e. x(t). It is this x(t) which gives a sense of past, present and future. A quantum system, however, is a statistical system much like V(x) a potential. It does not follow a particle from one x to another in space because the potential renders stochastic hits. It is only interested in the overall resonance which forms. “t” in exp(-iEt) is associated with this resonance and not with single particle motion. Thus there is no classical sense of past, present or future in the quantum bound state at least with respect to the single particle. An entire quantum system may, however, evolve, but a bound system simply cycles through time. The single particle has no time attributes except dt associated with 1/E where E=pp/2m. In part 1 we argued that p may be thought of in terms of impulse (hits). In such a case it is not directly linked to space and time. We also argued that p=mo v even in the quantum case so dx and dt must be present indirectly. In this note we revisit these ideas by following (1) and deriving quantum mechanical behaviour for a free particle from the classical Lagarangian equations dL/dx (partial) = p and H= pv - L. This directly shows that a quantum picture deals with dx and dt for a single free particle and so the idea of x(t) only holds in an rms sense. For a single free particle, matters differ from the bound case because p = mo vrms (free particle). Thus we argue that exp(ipx) must be viewed in terms of probability and is only manifest if an interaction occurs.Such interactions, however, may be continuously occurring as in the case of the Bohm Aharonov effect in which there is a magnetic vector potential existing in a large region of space.
Note: The equations d/dx partial [ T L] = p and d/dt partial [ TL] = -E hold for both the relativistic and nonrelativistic free particles because dL/dv = p and H=pv-L hold for both the relativistic and nonrelativistic free particle. In part I of this note, we argued that in quantum bound states one has a notion of dx and dt associated with p and E i.e. dx→1/p and dt→1/E, but not of continuous time associated with particle motion. exp(-iE t) contains time t, but this refers to a cycling time associated with various exp(ip1x) free particle probabilities combining with exp(ikx) to form exp(ipx). In other words there is a certain time resolution as well as a space resolution (wavelength) for the single particle and time t refers to the entire quantum resonance. In a bound system, there is also a vrms(x) which holds for every quantum energy state En and this is linked to a continuous space time picture i.e. x(t). It is this x(t) which gives a sense of past, present and future. A quantum system, however, is a statistical system much like V(x) a potential. It does not follow a particle from one x to another in space because the potential renders stochastic hits. It is only interested in the overall resonance which forms. “t” in exp(-iEt) is associated with this resonance and not with single particle motion. Thus there is no classical sense of past, present or future in the quantum bound state at least with respect to the single particle. An entire quantum system may, however, evolve, but a bound system simply cycles through time. The single particle has no time attributes except dt associated with 1/E where E=pp/2m. In part 1 we argued that p may be thought of in terms of impulse (hits). In such a case it is not directly linked to space and time. We also argued that p=mo v even in the quantum case so dx and dt must be present indirectly. In this note we revisit these ideas by following (1) and deriving quantum mechanical behaviour for a free particle from the classical Lagarangian equations dL/dx (partial) = p and H= pv - L. This directly shows that a quantum picture deals with dx and dt for a single free particle and so the idea of x(t) only holds in an rms sense. For a single free particle, matters differ from the bound case because p = mo vrms (free particle). Thus we argue that exp(ipx) must be viewed in terms of probability and is only manifest if an interaction occurs.Such interactions, however, may be continuously occurring as in the case of the Bohm Aharonov effect in which there is a magnetic vector potential existing in a large region of space.
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 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.
A classical bound system may be said to have a spatial density proportional to the amount of time dt a particle spends in a constant dx region i.e. C/v(x) (1) We stress that in this scenario there is a well defined and constant dx so that “dt” may be linked to velocity v(x). We also note that there is no spatial representation of action-reaction here even though this exists in Newton’s theory. In a classical Maxwell-Boltzmann gas there are many particles which each act dynamically according to .5mvv+V(x) = E (where v and E are particular to the particle when it is not colliding). At the same time there is a spatial density which “represents” macroscopic static chunks of matter which are balanced through pressure differences (P=spatial density(x) RT) and -dV/dx density where V(x) is the potential. From the pressure equation, T is proportional to energy which in turn is proportional to RMS speed squared. RMS speed may then be linked to a dt if a fixed dx is understood. In the classical statistical case, however, spatial density is directly linked to the number of particles in a region. Time enters in the pressure expression through a term proportional to the average kinetic energy i.e. pv where p is like an impulse and v a flux. What happens in a single particle quantum bound state? In a sense it is like a classical bound state as the equation: prms(x)prms(x)/2m + V(x) = En holds for any energy level. Can one then take vrms(x), choose a fixed dx and argue that spatial density should be equal to dt or C/vrms(x)? We argue that one cannot due to the lack of defined “dx” quantity. Why is dx problematic to define? We argue that it is because quantum mechanics is based on action-reaction x-probaiblity distributions linked to impulse. For example, exp(ipx) for a free particle captures only p or impulse (not velocity) and cos(px) and sin(px) are its action-reaction regions.These lead ultimately to a density is given by the well-known W*(x)W(x) where W(x)=Sum over p a(p)exp(ipx), although kinetic energy with pp/2m = .5pv is used to find W(x) as well. . W*(x)W(x) has crests and troughs similar to cos(px) in exp(ipx), except that these now have varying heights and widths. We try to argue that even though each x point maps to a different KEave(x), the hump physically represents action-reaction in the system (just as the humps in cos(px) do except for a single p) while KE(x) represents an average of p’s. This action-reaction is linked with impulses. In general, if one takes d/dx of the time-independent Schrodinger equation one finds an average impulse term, -dV/dx and KE(x) dW/dx / W where dW/dx / W is the average impulse. At peaks, dW/dx =0 and so an average force through time*impulse term balances -dV/dx=Newtonian force. These peaks are also associated with W*(x at peak)W(x at peak) which is the spatial density. We interpret this as representing the average time the single particle spends near x at peak. This value, however, follows from ideas of impulse and not from speed as there is no well defined dx even though there is a well defined vrms(x). Thus the ratio of W*(x)W(x) at two peaks is not the inverse of the ratio of vrms(x) except apparently when the energy level n becomes very high. We investigate these ideas.
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)).
Quantum mechanics is concerned with measurement and we suggest that in a bound problem impulses due to momentum are the form of measurement which occurs in x space. The Lorentz invariant A= -Et+px, which is also the free particle relativistic or nonrelativistic action with v=x/t, associates energy = pp/2m with time and p with x. Creating eigenvalue equations yields id/dt exp(-iEt) = E exp(-iEt) and -id/dx exp(ipx) = p exp(ipx). P (momentum) is associated with physical impulses and with its own two dimensional probability distribution exp(ipx). E is associated with t and has its own distribution in time exp(-iEt) which implies that the same E holds for each x point without any notion of x probability associated with it. In classical physics, a momentum p exists in a tiny dx (dx→0). For exp(ipx), a p impulse may occur within hbar/p with different probabilities, so one cannot restrict the measurement of p to a tiny dx unless dx is of the order of hbar/p. Nevertheless energy exists at all x points. Given that p measurements occur through specific impulses at different x points with exp(ipx) probabilities, one must calculate an average energy using {Sum over p pp/2m a(p) exp(ipx)} / {Sum over p a(p)exp(ipx)} using the probability associated with the measurement of p in space i.e. exp(ipx). This adds to V(x) to yield a constant E at each x point. We suggest that weighted exp(ipx)s are used to calculate average energy at each x, because it is through impulse measurements that one determines p at different x values and hence the average energy. One may note that one does not calculate {Sum over p a(p)exp(ipx) p } / {Sum over p a(p)exp(ipx)}. An impulse hit is associated with a certain probability and this measurement in x yields information about the corresponding energy pp/2m. Each impulse hit also yields information about a time probability exp(-i ppt/2m), but only if the particle is knocked out of the bound system i.e. only if pp/2m is the only energy present, which is not the case for a single particle bound system.
We argue that in a quantum bound state with one particle, the particle receives stochastic hits from a potential. As a result, if one examines a point x, the particle may be present at ti with momentum pi (in both the forward and backward directions) leading to a momentum distribution with a conditional probability P(p/x) = a(p) exp(ipx)/W(x) where W(x) is the wavefunction. Thus, one does not follow a stochastic particle in time, but rather describes it by a probability distribution at each x. One may next ask whether this conditional probability at x changes in time? This is not the actual time related to stochastic motion of the particle. To answer this question, we argue one must consider interactions which occur at x which are part of an energy equation. Even in classical physics, the energy equation at x indicates how a particle’s motion changes. Thus, we argue one does not simply take measurements at x of ti and pi, but that at each ti, a particle with pi is undergoing dynamic changes and these may be considered as well. If one describes the quantum picture by W(x) i.e. a momentum distribution, one may find a collective interaction given by -1/2m d/dx d/dx W + VW. The coupling between V(x)=Sum over k V(x)exp(ikx) and the particle conditional probabilities in W is apparent in the VW term. This energy term should describe whether the momentum distribution changes in time i.e should be proportional to dW/dt (partial derivative), but should also equal the average energy. Thus, a link exists between ensemble energy and time. In this note, we argue that these ideas may be applied to P(p/x) in a bound state resonance because the same E applies to each p.