In Part I we noted that in quantum mechanics, and even in the case of special relativity, there is an arbitrariness associated with x and t. For example, momentum in one dimension is exactly 0. One cannot arbitrarily shift it. If x=0, however, this is arbitrary, it may be shifted. We argued that this arbitrariness and the independence of x and t in a rest frame mo, x=0, t=to should carry over into the results seen in a moving frame, but in that case x’/t’=v because a Lorentz transformation links (x=0,t=0) to (x’=0, t’=0). Thus it appears that the arbitrariness of x,t and their independence have been broken, but we argue that a special Lorentz invariant -Et+px restores it by creating a time period hbar/E and wavelength hbar/p. In other words, a mathematical point is an abstraction representing a wavelength region defined by p through a complex probability exp(ipx) i.e. the wavefunction (and a similar exp(-iEt for time). In this note, we focus on these points in further detail. A point x is associated with p or -p in one dimension, but a distribution, say cos(px), combines a fixed p with probabilities and gives the impression of positive and negative momentum values when the slope is positive or negative. Thus cos(px) by itself shows no overall translational motion, but contains regions of positive or negative momentum i.e. zero point motion. One must superimpose on this zero point motion (which exists for all energies) actual “translation” to the right or left. Given that positive and negative p’s already exist in one dimension which have nothing to do with translation, one is forced to move into a second dimension to create the translational motion which is associated with a probability such that each x point has the same value. This then leads to exp(ipx). Next we consider a bound state situation. In the case of exp(ipx), cos(px) and sin(px) are linked to the positive/negative zero point type energy which has nothing to do with translation, while p has everything to do with translational motion. Thus if one is only interested in the translational motion (as Newton was) one may map exp(ipx) to p at x. An interesting feature, however, that appears in quantum mechanics is that is the zero point type motion which creates a very specific kind of averaging combines with pp/2m values to create the notion of translation(to the right and left at the same time so to speak because time is removed from the bound problem. This is very different from the quantum free particle which has p and pp/2m at each x point and is due to the two dimensional nature of exp(ipx) which leads to pexp(ipx) - pexp(-px) not equal 0. Thus the nature of “acceleration” in a bound quantum system is closely linked to zero point energy (i.e. the back and forth action-reaction momentum) which is already present in a free particle. In fact, for a bound system, this zero type motion is linked to the average -idW/dx / W (W=wavefunction) which is not the same as prms(x) obtained from -1/2m d/dx dW/dx /W.
In Part I we noted that in quantum mechanics, and even in the case of special relativity, there is an arbitrariness associated with x and t. For example, momentum in one dimension is exactly 0. One cannot arbitrarily shift it. If x=0, however, this is arbitrary, it may be shifted. We argued that this arbitrariness and the independence of x and t in a rest frame mo, x=0, t=to should carry over into the results seen in a moving frame, but in that case x’/t’=v because a Lorentz transformation links (x=0,t=0) to (x’=0, t’=0). Thus it appears that the arbitrariness of x,t and their independence have been broken, but we argue that a special Lorentz invariant -Et+px restores it by creating a time period hbar/E and wavelength hbar/p. In other words, a mathematical point is an abstraction representing a wavelength region defined by p through a complex probability exp(ipx) i.e. the wavefunction (and a similar exp(-iEt for time). In this note, we focus on these points in further detail. A point x is associated with p or -p in one dimension, but a distribution, say cos(px), combines a fixed p with probabilities and gives the impression of positive and negative momentum values when the slope is positive or negative. Thus cos(px) by itself shows no overall translational motion, but contains regions of positive or negative momentum i.e. zero point motion. One must superimpose on this zero point motion (which exists for all energies) actual “translation” to the right or left. Given that positive and negative p’s already exist in one dimension which have nothing to do with translation, one is forced to move into a second dimension to create the translational motion which is associated with a probability such that each x point has the same value. This then leads to exp(ipx). Next we consider a bound state situation. In the case of exp(ipx), cos(px) and sin(px) are linked to the positive/negative zero point type energy which has nothing to do with translation, while p has everything to do with translational motion. Thus if one is only interested in the translational motion (as Newton was) one may map exp(ipx) to p at x. An interesting feature, however, that appears in quantum mechanics is that is the zero point type motion which creates a very specific kind of averaging combines with pp/2m values to create the notion of translation(to the right and left at the same time so to speak because time is removed from the bound problem. This is very different from the quantum free particle which has p and pp/2m at each x point and is due to the two dimensional nature of exp(ipx) which leads to pexp(ipx) - pexp(-px) not equal 0. Thus the nature of “acceleration” in a bound quantum system is closely linked to zero point energy (i.e. the back and forth action-reaction momentum) which is already present in a free particle. In fact, for a bound system, this zero type motion is linked to the average -idW/dx / W (W=wavefunction) which is not the same as prms(x) obtained from -1/2m d/dx dW/dx /W.
In Part 3, we started with the notion of a free particle probability linked to energy and momentum conservation in two body Newtonian elastic scattering. In particular, we argued that one needed a probability which shows an equal weight for any outcome (ei,ej) and (pi,pj) (momentum vector) set which conserves energy and momentum. This led to a first guess of exp(iC1 E) and exp(iC2 p), but in Part 3, we argued the momentum and energy are defined through a Lorentz transformation of mo at rest and such a transformation must also affect x,t → x’,t’. We concluded one cannot create any probability based on p and E without also considering x and t which transform according to the same transformation. This then allows one to create a Lorentz invariant probability. Here we take an opposite approach, i.e. one which does not involve p and E a priori. It is well known from classical physics that P(x)dx=dx/L, where L is an arbitrary length, describes a uniform distribution for particles at rest. One may, however, view these particles from two frames, one moving with constant v and the other with constant -v. This would seemingly introduce probabilities P(x1’,t1’,v) and P(x2’,t2’,-v) with no notion of energy or momentum present. It would seem that the product of the two for an x1’=x2’ and t1’=t2’ should equal the rest frame result. This suggests an exp(f(x,t,v)) solution, but there is no reason to have an ever increasing or decreasing exponential and so exp(i f(x,t,v)) seems like a solution. There is, however, a problem, because given P(x,t,v=0), and such a solution is then not P(x)= 1/L unless P(x,t,v)=1, which we reject as it assumes a moving particle is identical to one at rest in terms of probability.. In other words, there are two probabilities for the rest frame, P(x)=1/L and P(x,t,v=0) = exp(i f(x,t,v=0). Given that exp(i f(x,t,v) is based on Lorentz frames, one would also expect probability to be conserved and not be created or destroyed simply because a particle with rest mass is viewed from a rest frame or one moving at constant speed. This seems to force one to use relativistic variables linked to a Lorentz invariant, namely E,p as well as x,t. As a result, in Part 3 we argued that one cannot consider a probability of p,E without considering x,t as well and here we suggest that one cannot even consider a probability of x which is conserved in different constant moving frames without introducing t,p and E. Thus, P(x,t,v=0) which seemed like a “P(x)” type of probability in the rest frame is really a different probability, namely one in time exp(i f(x, v=0, t)), i.e. . for this to be Lorentz invariant, one has exp(-iEt+ipx) = exp(-iEt) for any x. In a sense, this is like P(x)=1/L in that each x carries the same weight, but there is still the time variable and the probability actually changes in time unlike P(x)=1/L. This seems to lead to a paradox because the given exp(-imocct) one may formally argue that exp(-px)=1 for p=0 which is the x part of the probability, yet above we argued that in order to obtain this value one should use P(x1’,t1’v)P(x2’,t2’,-v) at x1’=x2’ and t1’=t2’. The difference between P(x)=1/L is that the time measurement has infinite resolution, while exp(-imocc t) implies that it does not because otherwise information about x is the same. Thus, special relativistic considerations lead to the notion of uncertainty in time (hbar/E) and space hbar/p, but in the Newtonian scenario, these do not exist. We argue that the question then becomes: Why would one wish to change from a special relativistically related probability exp(-iEt+ipx) which shows physical uncertainty units of time and space, to one which shows infinite resolution for x and t, through exp(-iEt+ipx)exp(iEt+ipx)? We suggest that exp(-iEt+ipx) or exp(ipx) in time-independent problems is a dynamic probability needed to describe interactions. One may see that cos(px) and sin(px) may have positive and negative values, implying the removal of probability from one place and its addition somewhere else. One the interaction result has been obtained, one wishes to measure spatial results, i.e.count particles in bins in space. There is no need for the interaction information and it may be formally removed using exp(iEt-ipx) exp(-iEt+ipx).
It was argued in (1) that Fermat’s least time principle for light (in a two dimensional (x,y) reflection or refraction scenario) minimizes time by actually varying it in terms of a spatially invariant variable i.e. x (i.e. taking d/dx) with the flat face of the mirror/medium along the x axis at y=0. This is equivalent to conserving momentum in the x direction. Thus d/dx is associated with (px) which may be associated with the Lorentz invariant -Et+p dot r. In other words, d/dx emerges as a momentum operator (up to a factor) and d/dt as an energy operator, yet both are generators in time and space linked with invariance of constant values i.e. conservation. Furthermore an eigenvalue equation: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt) automatically (by the nature of eigenvalue) leads to orthogonal eigenfunctions linked to conservation of energy and momentum. In special relativity rest mass mo represents a constant in time which may exist at any point x. In fact, it is often thought of as a point, for example a center of mass point. The actual object with mo, however, may be large with moving parts all within a “container”. What then does x,t in -Et+px mean for constant E,p? It seems that x,t are center-of-mass coordinates for the overall system. We argued in previous notes that 1/p is proportional to a wavelength so for p→0, mo may be at any x point with the same probability. Within the system with moving parts, however, there is motion and an x scale for each moving piece. For a single particle bound system, there is a single energy at each x which is like mo and a potential V(x) and KE(x). In (2) we argued that given the free particle relationship -EE+pp= -momo (c=1) one may retain a constant E value by replacing E with E-V(x) and pp with <pp> (and not ). How do the probabilities appear? They are linked to -Et+px with frequency hbar/E and wavelength hbar/p and are specifically manifested by exp(-iEt) and exp(ipx). For and <pp> we use spatial probabilities i.e. W(x) = Sum over p a(p)exp(ipx). Each exp(ipx) is linked to the operator -id/dx and invariance, but pp is linked with the operator -d/dx d/dx and one does not use an eigenfunction of this operator. Thus the notion of conservation of momentum is retained in the exp(ipx)s and their linear combination W(x) yields a function such that an average kinetic energy exists at each x point i.e KE(x)=KEclassical = -1/2m d/dx dW/dx / W. This is combined with a V(x) to ensure constant E at each x point i.e. create a kind of mo piece linked with id/dt. We argue that E being constant at each x is a statistical equilibrium type of result. Each exp(ipx) is associated with an average motion of x=p/m t and given that exp(ipx) and exp(-ipx) are included, there is also the notion of which is positive and negative in different regions and integrates to 0. prms(x) from <pp> becomes the classical p(x). Thus bound state quantum mechanics seems to be based on a free particle treatment which follows from the special relativistic Lorentz invariants -EE+pp=-momo and -Et+px = constant, directly linked to conservation ideas. Conservation of momentum of an internal bound particle is maintained through exp(ipx), while <pp> by itself is not linked with conservation as there is no eigenstate, but is combined with a V(x) to create a constant En linked with the entire system (and P=0) which then becomes associated with invariance in time and the generator id/dt. In this note, we try to argue that the notion of W*(x)W(x) (where W(x)=wavefunction= Sum over p a(p)exp(ipx)) is spatial density and P(p)=a*(p)a(p) based on spatial invariance and conservation of momentum for a single particle bound state even though there seems to be no spatial invariance or notion of conservation of momentum in W(x) which is simply a solution of the time-independent Schrodinger equation. One may move from a W(x) scenario (i.e. Schrodinger equation) to a W*(x)W(x) picture, by trying to “pick out” invariant (p only terms) using the orthogonality of the exp(ipx)s. Thus W*(x)W(x) (spatial density) emerges naturally in this approach.
It is sometimes stated in the literature (1) that quantum mechanics is based on conditional probabilities. If this is the case, one would expect such conditional probabilities to appear in a single particle bound state. A free quantum particle is associated with a probability to interact of exp(ipx) (linked to 2-slit interference, 1-D reflection-refraction at an n1-n2 junction etc), but the kinetic energy and momentum are pp/2m (nonrelativistic) and p at all x even though each x has a different phase according to exp(ipx). The momentum is p at x, but the probability for an impulse hit at x is not the same even though its modulus is. When different probabilities exp(ipx) add in an OR situation, this leads to a spatial pattern which no longer has a modulus of 1, i.e. translational invariance is broken. Such a pattern also occurs for a single particle bound state W(x)=Sum over p a(p)exp(ipx). W(x) can have 0 points, but the average kinetic energy -1/2m d/dx dW/dx / W is not 0 at such a point. One has, on average (a statistical calculation), no average interaction at such an x point. One might ask why this should have anything to do with average kinetic energy? If one writes a conservation law: KEave(x) + V(x) = 0, then one considers kinetic energy interacting with potential energy. If there is no interaction at some x point for a given p (not W(x) on average), then it should not be part of the conservation of energy at this point. One is only interested in kinetic energy which interacts with potential energy at each x point. Thus, a(p) exp(ipx) + a(-p) exp(-ipx) (with a(p)=a(-p)) may have a 0 point at some x and so it does not have an average interaction at such a point, i.e. pp/2m (a(p)exp(ipx) + a(-p)exp(-ipx)) = 0. As a result, in a bound state, one considers kinetic energy per p, -p multiplied by the average probability (p, -p). Different a(p)exp(ipx) + a(-p)exp(-ipx) (a(p)=a(-p)) weights in this average picture indicate the extent to which the particle interacts. This idea holds for a single p,-p pair, but one must normalize over all such pairs, leading to the idea of a conditional probability. Given that hbar/p must be “created physically” in space so to speak, one must have interaction probability which creates this physical situation even in cos(px)+ i sin(px). This leads to the idea of interaction probability being 0, positive, but <1, or even negative and it is the interaction which allows one to consider conservation of energy with a potential V(x) in the first place. As a result, one cannot include a full pp/2m if there is negative, 0 or only particle interaction. Thus, pp/2m is multiplied by 2 a(p)cos(px) /W(x) for a(p)=a(-p).
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 have argued in Parts 1 and 2, that the free particle quantum form exp(-iEt+ipx), which we call a complex probability, arises from special relativity in order to allow for conservation of energy and momentum (in an elastic collision). One may see from single particle bound states how exp(-iEt) plays the part of a binding energy. We have noted in Parts 1 and 2, that exp(-iEt+ipx) follows from the Lorentz invariant -Et+px, but that one does not consider the deterministic x=vt equation which applies to the center-of-mass, but rather treats t and x as independent variables which force the independence of E and p (two different interaction variables) and also force separate E and p conservation. The question becomes: How is it possible to have x,t play two such different roles in special relativity? Here we try to answer this question. We argue that special relativity is based on determinism, i.e. x=vt. One may create a Lorentz transformation for a particle with rest mass mo at x=0 with t, a running variable. As seen from a frame moving with -v, one has x’=vt’. In other words, a separate x’,t’ is associated with each v. In Newtonian mechanics, it is possible to have mov1 and mov2 in the same frame (i.e. a frame with a ruler and clock), but in special relativity, each mo in the rest frame maps to a different x1’,t’ and x2’,t2’. This, however, seems to lead to a contradiction. If one considers two particles p1,E1 and p2,E2 (one dimensional motion) which elastically collide, then before the collision, they may be thought of as a composite particle (for more than a single x,t point) with energy E1+E2 and momentum p1+p2. Given the validity of special relativity, we argue that one may associate a Lorentz invariant -(E1+E2)t + (p1+p2) x with this composite particle. Furthermore, any outcome Ei+Ej = E1+E2 and pi+pj = p1+p2 would satisfy this same invariant. At this point there is no contradiction. The contradiction arises if one considers a single t and x for a composite particle, while internally there exist t1’,x1’ and t2’, x2’. The composite particle, in terms of special relativity, is not compatible with each -E’t’+p’x. The energies and momenta, however, add for each particle (in conservation equations) and so we argue that one should have a relativistically compatible approach which allows one to link each particle to a composite particle and see conservation of E and p. We note that it is precisely x=vt for each particle which leads to x1’,t1’ and x2’,t2’ and so argue that if one gives up this notion of determinism for x and t and introduces independent x, t values (fluctuations), then one can indeed retain the -Et+px form for each particle and use it to create composite particles. This seems to give rise to an AND situation which does not exist for both p,x and E,t in the deterministic relativistic approach. In that approach, one can only think of p’s and E’s adding separately, without any linking of x’s and t’s. In the fluctuation approach, it is possible to have the same x and t fluctuation points for each particle. We argue that it is this kind of thinking which leads to the probabilistic form exp(-iEt+ipx) which still respects the Lorentz invariant form and at the same time allows for conservation of E,p with the same t,x fluctuation points allowed. x=vt does not allow one to combine x1’,t1’ and x2’,t2’ for a composite particle because these may coincide for only one x,t point and one wishes to describe a composite particle with an x,t which exists over more than one x,t. This is why we suggest the physical fluctuation scenario which exists together with x=vt. x=vt describes the deterministic motion of the center-of-mass, while exp(-iEt+ipx) describes probabilistic hits in x and energy in t. It is special relativity which forces one to associate time fluctuations with E and spatial ones with p. As a result, we argue that special relativity gives rise to both probabilistic and deterministic versions, both of which respect the form -Et+px. The deterministic version is linked with x=vt center-of-mass motion, while exp(-iEt+ipx) represents probabilistic appearances of E in t and p in x, i.e. there is a spatial probability for impulse hit delivery.
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).