In previous notes (1) we discussed the free particle classical action in the form A= -Et+px in which p,x and E and t are independent for the relative and nonrelativistic cases. This leads to a scenario in which one cannot follow a particle in time i.e. there does not exist an x(t) as in classical mechanics. Instead one may develop eigenvalue equations -i d/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = E exp(iA) for the free particle. The solution exp(-iEt+ipx) is similar to a classical wave (which is a different solution following form A=-Et+px) in that it describes a kind of probability resonance in all of space. If one introduces a potential V(x), this is localized and one needs to think in terms of impulse hit resonances like exp(ikx) creating V(x). In other words, both an ensemble of exp(ipx) representing the particle and V(x)=Sum over k Vk exp(ikx) are described in terms of this resonance type periodic probability which interferes to create local features. One might argue that this feature of V(x) drives the particle. In this note we wish to examine the coupled oscillator H= p1p1/2 + p2p2/2 + w1w1/2(x1x1+x2x2) + .5w2w2(x1-x2)(x1-x2) (with m=1). Transforming variables leads to an equation with a separable solution, but in y=x1+x2 and y1=x1-x2. In this note we try to examine this solution in terms of the above resonance picture and also suggest a possible overall Shannon’s entropy.
In previous notes (1) we discussed the free particle classical action in the form A= -Et+px in which p,x and E and t are independent for the relative and nonrelativistic cases. This leads to a scenario in which one cannot follow a particle in time i.e. there does not exist an x(t) as in classical mechanics. Instead one may develop eigenvalue equations -i d/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = E exp(iA) for the free particle. The solution exp(-iEt+ipx) is similar to a classical wave (which is a different solution following form A=-Et+px) in that it describes a kind of probability resonance in all of space. If one introduces a potential V(x), this is localized and one needs to think in terms of impulse hit resonances like exp(ikx) creating V(x). In other words, both an ensemble of exp(ipx) representing the particle and V(x)=Sum over k Vk exp(ikx) are described in terms of this resonance type periodic probability which interferes to create local features. One might argue that this feature of V(x) drives the particle. In this note we wish to examine the coupled oscillator H= p1p1/2 + p2p2/2 + w1w1/2(x1x1+x2x2) + .5w2w2(x1-x2)(x1-x2) (with m=1). Transforming variables leads to an equation with a separable solution, but in y=x1+x2 and y1=x1-x2. In this note we try to examine this solution in terms of the above resonance picture and also suggest a possible overall Shannon’s entropy.
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.
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 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.
A single particle bound wavefunction may be written as a Fourier series W(x)exp(-iEt) = exp(-iEt) Sum over p a(p) exp(ipx). We argued in previous notes (1) that exp(ipx) is a statistical free particle conditional probability whose modulus of 1 corresponds to P(x)=constant. For the case of a potential V(x), we argue it is stochastic and write it as: Sum over k Vk exp(ikx). The stochastic potential gives rise to a momentum distribution: P(p/x) = a(p) exp(ipx) / W(x) at each x. A resonance, however, seems to occur because writing W(x) and V(x) in the time-independent Schrodinger equation as Fourier series, yields: pp/2m a(p) + Sum over k Vk a(p-k) = E a(p) ((1)) Thus, a(p) is really a combination of other a(p-k) or, in other words, there is a correlation between different free particle momentum states. If such is the case for single particle interactions with a stochastic potential, we argue it is possible a correlation may exist for two particles. In other words, a stochastic potential may not necessarily produce two separate resonances for two quantum particles such as ((1)), but may instead form an overall resonance involving both momenta p1 and p2 “interacting or communicating” with Vk as in the case of entanglement. We attempt to investigate these ideas in this note.
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.
A classical particle is considered to be a point described by x(t) i.e. x and t are both a set of points. In a one dimensional problem, however, dx= v(x) dt so if all dx tend to zero, then dt’s are small, but don’t tend to zero in the same way. In other words dx may be considered small, but of a finite size. To have a particle within dx, it should have a probability distribution that goes to zero at the endpoints of dx i.e. one could have a repeated set of these for a constant velocity. This, however, contradicts P(x)= C1 dt = C1 dx/v(x). Thus it seems first that there are two kinds of probability distributions and that the set of distributions fitting into each dx is incomplete because if this set is described by a function F(x), this function should have the same magnitude at F(x+dx) for a free particle. This suggests a phase function linear in x i.e. exp(i C1 x). P(x) = C1dx/v(x) is based on dynamics and no dynamics appears in exp(i C1 x). It seems that dynamics should occur leading to the possibility of exp(i p x). (This , however, is ad hoc.) We call this a dynamic probability as opposed to P(x). In quantum mechanics it is called a wavefunction. If one considers the classical free action (relativistic or nonrelativistic) with x and t independent, then A= -Et + px (1) and one may create eigenvalue equations -id/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = E exp(iA) so exp(ipx) also appears more formally. In other words a fixed resolution periodic distribution requires the same shifted distribution (i.e. cos(px) and sin(px)) kept as separate dimensions i.e. “i” in order to create a dynamic probability which is a phase function which may be linked to P(x). This phase function exp(ipx), however, consists of two classical-like waves showing that in each dimension probability is “carved” out of troughs and added to crests. The two dimensions allow for the dynamic probability to be consistent with P(x)=constant for a free particle.