In Part 1, we tried to obtain free particle quantum mechanics by writing the LHS side of dp/dt = F as a gradient because the RHS F = - grad V(x) may be written in this manner for a conservative force. Here we argue that just as V(x) has physical meaning, p = grad partial (-Et+px) (found in (1)) should represent the physical potential created by a free particle. That does not change the fact that the particle center-of-mass moves as x=vt, but the interaction created by the particle should be described in terms of a potential and this was found to be -Et+px. From the point of view of a force system (e.g. a 2-slit system) interacting with the potential, one needs a potential created by the free particle. The potential -Et+px shows that an interaction at t,x has the same potential value as one at t+hbar/E and x+hbar/p. Given that this potential follows from dp/dt=Force, we suggest it should be taken seriously and that it indicates there is the possibility of interaction at two different x points x and x+hbar/p in a stochastic manner. In other words, a single free particle could interact with one slit or the other of a 2-slit system if the slit separation is about hbar/p. This still leaves two questions. First, is such a two fold interaction deterministic or probabilistic. We suggest probabilistic because there are ranges for x and t. Second, why does the usual Lagrangian method of finding a stationary solution from d Integral dt L(x, dx/dt, t) = 0 (leading to d/dt dL/dv partial - dL/dx partial = 0) not show this range in delta x? We suggest that writing p = grad partial (-Et+px) (so the potential is Lorentz invariant) removes the dynamics of the particle from the interacting force system (2-slits). The 2-slit is interacting with a free particle potential (not with its dynamic motion. The two need not be the same. We suggest that there are two theories for describing a particle interaction, the single particle potential theory and Lagrangian formalism. We thus suggest that they must be linked and try to show how they compare and how they differ in a systematic manner.
In Part 1, we tried to obtain free particle quantum mechanics by writing the LHS side of dp/dt = F as a gradient because the RHS F = - grad V(x) may be written in this manner for a conservative force. Here we argue that just as V(x) has physical meaning, p = grad partial (-Et+px) (found in (1)) should represent the physical potential created by a free particle. That does not change the fact that the particle center-of-mass moves as x=vt, but the interaction created by the particle should be described in terms of a potential and this was found to be -Et+px. From the point of view of a force system (e.g. a 2-slit system) interacting with the potential, one needs a potential created by the free particle. The potential -Et+px shows that an interaction at t,x has the same potential value as one at t+hbar/E and x+hbar/p. Given that this potential follows from dp/dt=Force, we suggest it should be taken seriously and that it indicates there is the possibility of interaction at two different x points x and x+hbar/p in a stochastic manner. In other words, a single free particle could interact with one slit or the other of a 2-slit system if the slit separation is about hbar/p. This still leaves two questions. First, is such a two fold interaction deterministic or probabilistic. We suggest probabilistic because there are ranges for x and t. Second, why does the usual Lagrangian method of finding a stationary solution from d Integral dt L(x, dx/dt, t) = 0 (leading to d/dt dL/dv partial - dL/dx partial = 0) not show this range in delta x? We suggest that writing p = grad partial (-Et+px) (so the potential is Lorentz invariant) removes the dynamics of the particle from the interacting force system (2-slits). The 2-slit is interacting with a free particle potential (not with its dynamic motion. The two need not be the same. We suggest that there are two theories for describing a particle interaction, the single particle potential theory and Lagrangian formalism. We thus suggest that they must be linked and try to show how they compare and how they differ in a systematic manner.
In (1), it is argued that the Schrodinger (nonrelativistic) quantum equation may be developed from steady state type classical equations linked to the continuity equation and Newton’s second law. Given the nonrelativistic treatment of (1), F=dp/dt = d/dt (mv) = m dv/dt partial + mv dv/dx. In (1) an equation (III.2) for many particles is obtained from the equation for a single particle: Md (dv/dt partial + v dot grad v) = dF + hbarhbar/4m grad dot (grad grad d - 1/d (grad d) (grad d) ). In (1), it is further assumed that v = grad (phi) for conservative forces, where phi is a function of x,t. One may then see that for d=1 case, for a single particle, that phi= -Et+px and that one may write a Schrodinger equation which is essentially a conservation of energy equation. Here, we consider starting with dp/dt=Force and using a special relativistic treatment. We note that for a conservative force, F = -grad V and suggest trying to write the LHS in terms of a gradient for a free particle so the gradient may be “pulled out” from both sides of dp/dt=F. From a physical point of view, F is a pull or push, but we argue that V (potential) is also physical and that given that p is almost like a force, that there should be meaning to p= grad phi(x,t). This is the basis premise. In the case, F=0, we show that one may obtain -EE + p dot p = constant via this approach which involves writing p = d/dx (-Et+px) (one dimension). We note that associating dp/dt = F with a Lorentz invariant -Et+px shows the close association of Newton’s second law with special relativity because dp/dt=Force is relativistic as long as p=mov/sqrt(1-vv/cc). We suggest that this approach associates operators id/dt partial and -i grad partial (hbar=1) with E and p and allows one to write a free particle equation (Klein-Gordon) in terms of exp(-iEt+ipx), which is a quantum form. We suggest that one way in which to see free particle quantum mechanics emerge (albeit in the simplest case-that of a free relativistic particle) is to try to consider dp/dt in terms of a gradient suggesting that there may be x, t features even for E,p constant. In Newtonian mechanics, a constant E and p immediately implies that derivatives are 0. This need not be the case if one considers p = d/dx (-Et+px). The legitimacy of considering p (which is closely linked to a force) is linked to the legitimacy of considering F = -grad V. In other words, V is a potential which is another way of considering the features of a force and p = d/dx (-Et+px) may mean that -Et+px (and exp(-iEt+ipx) may be another way of considering p,E at a deeper level than simply stating that one has a constant E,p for a free particle. We argue that one does not need to consider a continuity equation as done in (1) and that the full approach follows from analysis of a single free particle (constant E,p) in the special relativistic case, which then yields also the nonrelativistic one.
In Part 1, we argued that P(x)dx=dx/L (where L is an arbitrary length) applies to a particle at rest. If one views such a particle from a frame moving at constant -v, it is moving. We argued that in such a case, the relevant probability is not simply linked to the position x, but also to the momentum p at x because this delivers an impulse hit. In particular, probability is presumably to be used in a physical problem with interactions and probability and a dynamical particle requires a dynamical probability describing both x,t and E,p. This led to exp(-iEt+px) which actually predicts intervals in space hbar/p and time hbar/E. In other words, a probabilistic view point leads to these results. Here, we ask the question: What if one simply wishes to use deterministic physics to describe a moving particle? After all, Newtonian mechanics is deterministic and one does not really need to focus on P(x)=1/L or its modification for a moving particle. In such a case, a particle moving with constant v (starting at x=0, t=0) is described by x/t=v both in Newtonian mechanics and special relativity. There is no uncertainty in x and t here which begs the question: Why does it appear in the probabilistic formalism of Part 1? In particular, exp(-iEt+ipx) derived in Part 1 implies physical regions: dx = hbar/p and dt=hbar/E and these seem to be absent in x/t=v and this is a problem. We suggest that a probabilistic approach is buried in the Lagrangian formalism which leads to the result d/dt v = 0, i.e describes the particle’s motion in space. This motion is actually x/t=v, but this equation only presents a subset of the description of a particle that is moving with constant speed, even in a deterministic Lagrangian/Action approach. We have already pointed this out in (1). We briefly explain it again here, but the real question is why does one need to consider the Lagrangian in order to describe x/t=v? In other words, the probabilistic approach associated with v=x/t only emerges in the Lagrangian formalism using a relativistic action Lt = -Et+px, with x/t=v. Ignoring the action and Lagrangian leads to an absence of hbar/p and hbar/E and one is only concerned with x/t=v. We note that the Lagrangian is primarily linked to the result: p = dL/dv partial and so argue that one is not simply interested in motion (x/t=v), but also in interaction because p is associated with an impulse hit. Thus, the Lagrangian approach is more than simply a mechanism which describes x,t motion, it is associated with p and E (through Hamiltonian = E = pv-L.) Using the Lagrangian approach suggests that it does not suffice to consider x and t by themselves when p and E are present. One must consider all four variables x,t,E,p which is the same point made in Part 1. The physics of the problem dictates that it is not enough to simply think in terms of x,t,v. The Lagrangian approach, which is constructed to ultimately yield x/t=v, when written in terms of E, p, i.e. Lt = -Et+px demonstrates uncertainty in x, hbar/p and t, hbar/E as shown in (1). This uncertainty must then be written in terms of a probability which is Lorentz invariant and leads to exp(-iEt+ipx). The point is that one would not even consider a probability if one did not consider E,p and x,t together as a complete treatment of a moving particle. In other words, x/t=v is an incomplete description of the state of a particle moving with constant speed, even in a deterministic framework.. At first this might seem surprising, but it is well known that energy and p are linked to interactions. In special relativity, one does not consider x,t,v alone and then compute p and E from these, but considers four vectors and Lorentz invariant equations. It is possible to create a Lorentz invariant equation involving all four state variables x,t,E,p and it turns out that this is the relativistic Lagrangian which describes motion, so this approach is more general than simply stating x/t=v. Thus, delta x = hbar/p and delta t= hbar/E appear in both the probabilistic approach of Part 1 and the deterministic one of the Lagrangian used here.
We try to consider free particle and single bound state quantum mechanics from the viewpoint of classical physics. We first assume that a free particle moves as x=vt. Secondly, we assume that dx and dt cannot be 0 as in Newtonian mechanics, but must be finite. This begs the question: How does one measure dx and dt? If one can measure a particle at a given x and t, then dx→0, dt→0. Thus, finite dx, dt means that a given p=constant and pp/2m=kinetic energy (nonrelativistic) interact with a force over a finite dx. In other words, one does not know where the interaction occurs within dx. As long as the free particle moves without interacting, then x=vt and there is no interest in any dx or interactions. For example, one may measure very large x and t ranges. Even if there are dx and dt effects at the start and end points, these are small compared to the large x and t values. As a result, one has a F(x,p) probability within dx, which repeats itself for the next dx (i.e. is periodic), where F is a probability for a p impulse hit at x within dx. At the same time, the particle moves with x=vt it seems within dx. In general, one might then think that F(x,p) and x=vt are decoupled. In previous notes, we have argued that one way to couple F(x,p) and x=vt is through special relativity, i.e. A = -Et+px. X and t values used would presumably satisfy x=vt, but for any x,t one has x+hbar/p and t+hbar/E as solutions yielding the same A value and so might consider dx= hbar/p and dt=hbar/E. The question we ask here is whether one may arrive at this conclusion without using special relativity. The answer seems to be that one can if one suggests that F(x,p) must contain information making it consistent with x=vt, i.e. with every x and t point having the same weight. Clearly, however, each x,t do not have the same weight within dx and dt, but are rather governed by F(x,p). This means, however, that F(x,-p) also represents a probability distribution and so if one has F(x,p)F(x,-p) there is no momentum at a given x point. In such a case, the distribution within dx must disappear and one must have a uniform distribution. This holds even if F(x,p) has the same dx value for any p. Thus, F(x,p)F(x,-p) = constant and must be periodic in x. This implies that F(x,p) = 1/F(x,-p) for constant=1. Given that F(x,p) =0 at the initial and final points of dx, then F(x,-p) = infinite at these points and that is not allowable. Thus, one is forced to use complex variables and try exp(i g(p) x). If g(p) is a function of p (i.e. not a constant), then conservation of momentum implies that exp(i g(p1) x) exp(i g(p2) x) = exp(i g(p1+p2) x), suggesting exp(ipx). What, however, if g(p) = constant? In such a case, one has issues creating an average kinetic energy: ( Sum over p a(p) exp(i constant x) pp/2m) / (Sum over p a(p) exp(i constant x) ((1)) This leads to a kinetic energy which does not depend on x and cannot be used with V(x) in a conservation of energy equation. Thus, we choose exp(ipx). Note: there is no reason to choose a(p,x) because the physics of finite dx and dt is restricted to exp(i constant x). Finally, we try to explain why the same dx=hbar/p and dt=hbar/E together with x=vt follow from special relativity.
We have argued in previous notes that free particle quantum mechanics arises from special relativity and is dependent on both the deterministic x=vt and fluctuation/probabilistic delta x= hbar/p and delta t = hbar/E. In other words, both determinism and probability are inherent to special relativity and it is also special relativity which defines energy and momentum. At first, it may seem that these are rather strong suggestions. We argue, however, that perhaps these ideas are already present in classical Lagrangian/Stationary Action theory. This theory was developed to describe Newtonian mechanics, but we argue that because it uses the notions of E (energy) and p (momentum), it must be linked to special relativity which yields the full expressions for these functions. The interesting feature of stationary Lagrangian theory is that a notion of fluctuations in path (x,v,t) are used to ultimately obtain the deterministic path x=vt for a free particle as the stationary one. The notion of a stationary, deterministic path x=vt is present, but so is the notion of fluctuations about this path. The problem is that the stationary action process is a mathematically one and does not attribute and physical features to the fluctuations about the stationary path, whereas we argue that these fluctuations are linked with probabilistic impulse hits or E interactions in time and are directly associated with the quantum wavefunction (complex probability) exp(-iEt+ipx) which physically manifests itself (as in 2-slit experiments etc). In order to connect stationary classical action theory to special relativity, we use H = pv - L -> -Hdt + pdx = L dt. If one considers fluctuations of this equation, the RHS set to 0 leads to the stationary deterministic x=vt, while the LHS leads to delta t = hbar/E and delta x = hbar/p. In other words, there is a delta x and delta t not linked to v = delta x / delta t. We suggest that this approach shows both probabilistic and deterministic features present side by side and is directly linked to special relativity because -Hdt + pdx is of the form of the Lorentz invariant -Et+px.
Classically, probability is considered in terms of the number of possible outcomes. For example, a die toss has six possible outcomes and a coin toss, two. Probability calculations make use of probabilities and hence of the total number of possible outcomes. Thus, they are not deterministic calculations which follow “what happens” as in Newtonian mechanics. We suggest that if free particle quantum mechanics is probabilistic, it must also involve the notion of possible outcomes in an experiment and not represent a deterministic approach. We suggest, however, that something interesting applies to quantum mechanics because we have suggested that free particle quantum mechanics arises from the Lorentz invariant A = -Et+px. We note that there exist x,t trajectory points which follow x=vt. We also note, however, that one may introduce the idea of intervals dx, dt (because Newton’s dx→0, dt→0 is an idealization) and that A is unchanged for x,t and x+hbar/p , t+hbar/E. In other words, we suggest that there are intervals with probability about each x,t trajectory point. A trajectory point is associated with deterministic physics, but probability about this point is not. In other words, one has a kind of hybrid probability scheme. Thus, exp(ipx) and exp(-iEt) really represent fluctuation probabilities about a trajectory point x and t. These ranges of possible x and t fluctuations about a trajectory point x,t allow for further probability to occur in certain interactions. We consider two examples. The first is 2-slit interference which involves a deterministic particle moving along a straight line towards a 2-slit mechanism perpendicular to its path. At the 2-slit apparatus, one considers a probabilistic calculation as the uncertainty in x, i.e. dx=hbar/p allows for possible interaction with both slits. This involves considering all possible outcomes (i.e. motion through one slit or the other which is a classical situation), but exp(i p dot r1) and exp(i p dot r2) “sit on top” of these trajectories at the 2-slit apparatus and one has interference of the probabilities linked with each classical possible outcome. Thus, there are two sets of probabilities which are intertwined, the probability range hbar/p for each path and the two possible paths which have equal probability. One must consider the possible outcomes for both sets together. We suggest that a similar analysis applies to the case of 1-D reflection-refraction for x=0 at an n1-n2 index of refraction junction. Thus, we suggest that a deterministic free particle with x=vt may encounter probabilistic interaction situations, such as a two-slit apparatus or an n1-n2 index of refraction junction. These are probabilistic only because there is a probability fluctuation scheme in x associated with each deterministic x point. It is this probability which allows for a second probability to arise due to interaction with the slits or at the 1-D n1-n2 junction. In such a case, one must consider all possible outcomes which means that one does not only consider the exp(ipx) type probabilities, but also those linked with the physical interaction, i.e. two possible paths, one from each slit, or an incident, reflected and refracted photon at x=0 (n1-n2 junction). It is a combination of all possible paths which must be used in calculations. We thus suggest that even though there is a sense of determinism, i..e free particles move as x=vt and these exist in the 2-slit case and the 1-D reflection-refraction case, in a probabilistic calculation one must consider all probabilistic outcomes consistent with these. Thus, quantum mechanical calculations are true probabilistic calculations in the sense that they involve all possible outcomes in an analysis.
In Part 1, we argued that Newtonian mechanics essentially contains the notion of probability found in the free particle wavefunction exp(-iEt+ipx). The difference is that the wavefunction/probability quantifies dx=hbar/p and dt=hbar/E. The standard Newtonian approach is to have dx->0 dt->0 in order not to deal with the lengths of these intervals. This limiting feature may work in the classical world (but not for photons reflecting and refracting through windows), as the actual sizes of dx and dt may in general be very small in the classical world, but they are not small in the case of atomic or nuclear lengths. The question is: In Newtonian mechanics, without any reference to exp(-iEt+ipx), how would one obtain finite values for dx and dt? Furthermore, why should one wish to obtain dx,dt values associated with x,t in x=vt. We suggest that because one deals with a moving object, and not a stationary one, any x,t on a x=vt trajectory should be associated with a dx,dt to verify its existence. Unlike the stationary case, for which one may strike the particle and make it move, dx,dt for the moving particle should be dynamical and associated with Lorentz invariance. We suggest that it is meaningless to speak of x,t for a moving particle without a corresponding dx,dt. If one can obtain dx,dt values, then one can argue, as we did in Part 1, that Newtonian mechanics contains the notion of an uncertainty dx about x and dt about t for a free particle. This is essentially the uncertainty which allows the 2-slit interference to occur and solves the probabilistic reflection-refraction problem at an n1-n2 index of refraction interface. Of course, if probability must exist within dx and dt, one requires a math expression for such a probability (i.e. exp(-iEt+ipx). One cannot simply use Newtonian mechanics because it does not make use of such a probability, but at least it predicts that such a probability should exist, we argue. Here we show that one may directly use special relativity to show that dx=hbar/p and dt=hbar/E in Newtonian mechanics, such that one has a dx and dt associated with the same value of the classical action as x,t for which x=vt. The classical action describes, for various x,t values describes the x(t)) trajectory in spacetime and so having built-in dx,dt values in this same action ( which is the Lorentz invariant -Et+px both nonrelativistically and relativistically) shows that Newtonian mechanics must be associated with finite dx,dt. The reason that we consider a dx and dt in the first place is that v=dx/dt. The Newtonian approach is to take the limit dx->0 and dt->0. This, we argue, is associated with a coarse-grained equation x=vt which holds both relativistically and nonrelativistically. This is associated with a coarse-grained classical action.Each x,t on the trajectory x=vt yields a different action value for the action (x/v*.5mvv or -x/v mo sqrt(1-vv/cc) or -Et+px where x/t=v. One may consider x=vt and x+dx1 and t+dt1 as the next points in the trajectory. According to Newtonian mechanics, one may make these values arbitrarily small. Furthermore, x=vt holds both nonrelativsitically and nonrelativisitically. How then can one propose a finite valued dx and dt? We argue that the manner in which this may be achieved is to have a dx,dt associated with x,t which does not change the action value for x,t, i.e. does not change -Et+px. This set dx,dt is finite valued even though the next x,t considered may have a dx1, dt1 value such that dx1<<dx and dt1<<dt. If, however, dx and dt are to be taken seriously, the dx1 and dt1 less than dx,dt are simply math values, not physical ones. In other words, one needs a dx,dt to justify the existence of x,t and if x=vt allows any x+dx1 (dx1<<hbar/p), then this is a coarse-grained equation. The coarse-grained action, however, allows for the defining dx=hbar/p and dt=hbar/E even if the action itself is coarse-grained.