Classical Newtonian mechanics separates a particle (effect) from a force (cause). A particle has properties at a given x (and time t), but if it interacts with a force, this interaction occurs over dt and dx. Given that many forces do not change in time, we argue that a notion of a force is inseparable from the idea of dx. If one gives the value of a force at x, this value acts within a dx, which in Newtonian mechanics tends to 0. We have argued before that a particle with momentum represents a force or impulse itself, but at the same time may receive impulse hits from other sources (e.g. a Newtonian force -dV/dx). If a particle represents a force(impulse) it should be inseparable from the notion of dx, even if it has a constant p (momentum) and is traveling through space. Thus the idea of a momentum defining an internal ruler gradation through a probability distribution in x which we have presented in previous notes is compatible with the idea of describing a particle as a force. If one only considers a particle as an effect, there would be no need to worry about dx until a force(x) appears. An internal dx is not defined by an external ruler and does not need to tend to 0 as in Newtonian mechanics. Rather it is statistical in nature as p is a fixed impulse, but the probability to be at different x points within dx (called a wavelength) differ. In (1) we compared a classical Maxwell-Boltzmann gas to a quantum free particle. We argued that d/d density Pressure = RT = average pressure of a particle. We suggested this is analogous to the quantum free particle situation: (-id/dx probability(p,x) ) / probability(p,x) = p where probability(p,x) is unnormalized. In other words a change in a probability yields a deterministic value p. The notion of a deterministic value p associated with a probability distribution in what would be another deterministic variable x in Newtonian mechanics (except now one has a dx distribution because the particle is a force/impulse) has consequences. If one wishes to have a particle sit at x=0, say in an oscillator system, one needs to use a wavelength which is fairly small and centered at x=0. The wavelength, however, is associated with momentum and even being small spreads out from x=0 meaning there is potential energy .5kxx as well. Thus one may use an ensemble of forces/impulses Sum over p a(p)exp(ipx) to create a “dx” about x=0, but at the same time this ensemble represents an average particle (defined at x with prms(x)) which must satisfy KE(x) + V(x) = En i.e. the classical conservation expression. In other words, one sees again that average statistical quantities may behave as deterministic ones.
Classical Newtonian mechanics separates a particle (effect) from a force (cause). A particle has properties at a given x (and time t), but if it interacts with a force, this interaction occurs over dt and dx. Given that many forces do not change in time, we argue that a notion of a force is inseparable from the idea of dx. If one gives the value of a force at x, this value acts within a dx, which in Newtonian mechanics tends to 0. We have argued before that a particle with momentum represents a force or impulse itself, but at the same time may receive impulse hits from other sources (e.g. a Newtonian force -dV/dx). If a particle represents a force(impulse) it should be inseparable from the notion of dx, even if it has a constant p (momentum) and is traveling through space. Thus the idea of a momentum defining an internal ruler gradation through a probability distribution in x which we have presented in previous notes is compatible with the idea of describing a particle as a force. If one only considers a particle as an effect, there would be no need to worry about dx until a force(x) appears. An internal dx is not defined by an external ruler and does not need to tend to 0 as in Newtonian mechanics. Rather it is statistical in nature as p is a fixed impulse, but the probability to be at different x points within dx (called a wavelength) differ. In (1) we compared a classical Maxwell-Boltzmann gas to a quantum free particle. We argued that d/d density Pressure = RT = average pressure of a particle. We suggested this is analogous to the quantum free particle situation: (-id/dx probability(p,x) ) / probability(p,x) = p where probability(p,x) is unnormalized. In other words a change in a probability yields a deterministic value p. The notion of a deterministic value p associated with a probability distribution in what would be another deterministic variable x in Newtonian mechanics (except now one has a dx distribution because the particle is a force/impulse) has consequences. If one wishes to have a particle sit at x=0, say in an oscillator system, one needs to use a wavelength which is fairly small and centered at x=0. The wavelength, however, is associated with momentum and even being small spreads out from x=0 meaning there is potential energy .5kxx as well. Thus one may use an ensemble of forces/impulses Sum over p a(p)exp(ipx) to create a “dx” about x=0, but at the same time this ensemble represents an average particle (defined at x with prms(x)) which must satisfy KE(x) + V(x) = En i.e. the classical conservation expression. In other words, one sees again that average statistical quantities may behave as deterministic ones.
Newtonian force is based on a change in momentum in time, but momentum is a vector and so may change in magnitude and direction. In many cases, the change is of one form or the other. For example, for a bound particle the change is in kinetic energy i.e. in magnitude, while for a particle moving at constant speed in a circle or pressure in a Maxwell-Boltzmann gas, the change is in direction. For an MB gas, the probability for p and -p are the same so the average momentum at any point x is 0 and thus chunks of the gas do not move, even though a single particle in the gas moves. The pressure, on the other hand, is the average of 2pv (2p for impulse, v for flux) and so pressure for p and -p do not cancel. A quantum free particle is based on exp(ipx) which does not contain time and is a dynamical probability, This is both an eigenfunction of the momentum operator -id/dx and the kinetic energy operator -1/2m d/dx d/dx, yet as pointed out in Part II, there are two pieces (cos(px),sin(px)) with a spatial density and hence a kind of pressure/force. These, however, combine to give the sense of constant motion in one direction. We argue this is due to a dynamical probability, unlike a static balance of pressures/forces found in an MB gas or Newtonian statics. There is a density with pressure distribution cos(px), but this is continuously changing because the particle moves in one direction with constant p. Thus the two pressure distributions “balance” so to speak to create a constant p. Interference of exp(ipx) and say exp(-ipx) may “bring out” a “static” spatial density i.e. 2cos(px), but behind the scene the dynamical probabilities still exist. It is these which “hold” the cos(px) together which appears strange from a static point of view. For example, a quantum bound state has a static W*(x)W(x) spatial density at the same time that the single particle has an average classical kinetic energy KE(x)=En-V(x). Interference also gives rise to the notion of an impulse-pressure in quantum as one ignores the isin(px) and i sin(-px) terms. Creating a pressure scenario with p and -p leads to two considerations. First, there is an impulse-pressure (defined without flux) at each x point and secondly, this impulse-pressure changes with x. The time-independent Schrodinger equation is formulated in terms of energy (i.e. a scalar): KE(x)+V(x)=En where KE(x)= {-1/2m d/dx dW/dx} \ W and W=Sum over p a(p)exp(ipx). Thus, as in Newtonian mechanics, one thinks of -dV/dx solely accelerating a particle. The quantum particle however, is described by an ensemble W(x) which has pressure. This pressure is due to a combination of the form exp(ipx) and V(x), so V(x) does not simply change root mean square speed. exp(ipx), by itself, creates pressure when interference occurs as in a two slit scenario. We try to examine these ideas in this note.
Newton’s second law dp/dt = Force seems to be written in terms of a cause F(x) and effect p(x), because momentum is written in terms of a change i.e. d/dt. His third law, however,describes two causes i.e equal and opposite causes (forces) leading to two effects. We argue that a constant momentum, the effect of a force, may also act as a “force” if divided by a constant time i.e. change in p / dt(constant) = Force. A conservative external force may be written in terms of the derivative of a potential i.e. -dV/dx, but we suggest that a quantum free particle is linked with two distributions cos(px), sin(px) which describe an internal force/acceleration for even a constant p. The two combine in two dimensions i.e. exp(ipx) to create a type of probability which has a constant modulus i..e treats each x point equally. Each x point is treated equally because there is no external force -dV/dx, but we argue that there is still an internal one linked with probability and spatial distributions i.e cos(px),sin(px). If an impulse is applied to a quantum free particle to change it from p1 to p2, the change in energy is completely described in terms of momentum i..e p2p2/2m - p1p2/2m. There is no need to consider space, but at the same time there is a change in the two distributions cos(px) and sin(px) due to the change in p. Thus a constant p is linked with two spatial distributions which have physical consequences i.e. two slit interference etc. There is an internal distribution in space. If a quantum particle is placed in a potential V(x), there is now both an external potential, but also an internal one associated with each exp(ipx) if one assumes an ensemble of exp(ipx)’s is created i.e. W(x)=wavefunction = Sum over p a(p)exp(ipx). The point we make is that both the external and internal potentials are linked with x distributions, thus applying d/dx effects both, but Newton’s conservation of energy KE(x) + V(x) = E deals with changes in kinetic energy only linked to the external potential V(x). Formally, one may write: -1/2m d/dx dW/dx / W + V(x) + E, but we argue that external and internal potentials “interact” and we try to describe the situation considering V(x)W(x) to demonstrate the interplay of internal and external forces from the point of view of the internal force (probability) exp(ipx) and statistical mechanical loss of information i.e. equilibrium.
In Part III we considered the quantum free particle wavefunction exp(ipx) as consisting of two parts cos(px) and sin(px) treated as two different dimensions. We also argued that each, is a probability distribution in space and so associated with an internal force. One may notice that exp(ipx) does not contain time, but still predicts the direction of motion. Given that one cannot consider x(t) within a wavelength hbar/p even though on average the particle moves as x=p/m t, there is no real sense of time within a wavelength. Extending this idea, one repeats cos(px) humps. Given that the particle is moving, one must somehow juxtapose two spatial probability densities. The derivative of the distribution is proportional to average momentum and so one may see the notion of action-reaction occurring between cos(px) and sin(px) and within cos(px) and sin(px) accounting for the sign changes with x. We argue that d/dx cos(px) regions which change sign imply that the impulse direction changes. Furthermore, cos(px) has regions near the endpoints of the wavelength which have very low values (close to zero) which is fine if one has two juxtapositioned distributions. If, however, one creates a single particle bound state, exp(ipx) and exp(-ipx) for a(p)=a(-p) (a specific parity) give the appearance of a cancelation leaving the impression of 2cos(px) i.e a single “static” distribution. This occurs because there is no internal time in the picture and so p and -p probabilities add. The so-called canceled pieces have not really disappeared. as will be argued, they simply do not contribute to an average impulse which is what experiments apparently detect. Thus “negative” quantum probability is linked to impulse in the opposite direction, we argue. Thus interference is really linked to the notion of probabilistic impulses i.e. p exp(ipx) even though exp(ipx) is used as a kind of probability. exp(ipx) contains internal action-reaction impulse hits which have experimental consequences as we argue is shown in a reflection/refraction problem, but are also linked with impulse hits associated with p. Thus there are two sets of forces to deal with. This, we argue, explains why one writes the energy conservation equation AA/c1=BB/c1+CC/c2 for reflection/refraction where A,B,C represent the incident, reflected and refracted rays and AA simply shows that intensity/flux is positive as two equations: A+B=C and Ap - pB = p2C with E=pc. The two equations,, needed to solve for B,C in terms of A,, describe the internal force in exp(ipx) and the probabilistic impulse p exp(ipx). For a single particle bound state, one considers a linear combination of free particle states indicating that impulse hits represent the underlying physics. Given the apparent spatial invariance of a free particle exp(ipx), one may wonder how a kinetic energy profile appears. We argue that different p values yield different peak points for maximum impulse hits and these are out of synch leading to a spatial distribution of KE(x) or d/dx KE(x) which is equivalent to -dV/dx =Force where V(x) is the potential.
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.
We have argued in previous notes that the quantum free particle wavefunction exp(ipx) represents two spatial distributions in two dimensions (i.e. at different times). We have further argued that a spatial distribution is linked to p, the impulse, needed to create the momentum in the first place. Thus a free particle contains a kind of stored impulse/force manifested by the distributions cos(px) and sin(px). Two dimensions are needed to give the semblance of smooth motion i.e. x=p/m t on average. In (1) we argued that the even in the case of a Newtonian impulse Integral F dt which one might assume occurs at a point x (as in two body collisions in a Maxwell-Boltzmann gas), there must be a dx range as the energy pp/2m follows from Integral Fdx which requires such a dx. In Newtonian mechanics everything is smooth, so there is constant acceleration within dx, but we argue that such a smooth picture violates the idea of action-reaction and inertia. Thus we argue that Newtonian mechanics should lead to the notion of an x-distribution within dx even if the particle is a point. This x-distribution which exists as the particle is under force does not disappear when the external force is removed because a particle with momentum p represents an impulse (which in turn includes a force). With the external force removed, one might expect there is nothing to maintain an x-distribution i.e. action/reaction has stopped. In one sense this is true as the particle moves through space as if each x point carries the same weight. On the other hand, the Newtonian idea of inertia, even with no momentum suggests one does not simply have a translating object. The idea of inertia i.e. rest mass should be linked then to the spatial distribution associated with the wavelength hbar/p which we argue happens in special relativity.
In Part I, we considered the ideal gas law P= density(x) RT of a Maxwell-Botlzmann gas, which follows from the notion of impulse 2p multiplied by v (flux) and then averaged,and examined how it balanced (on two sides of a box) a force -dV(x)/dx multiplied by density. We argued that in a quantum bound state there is no notion of internal time and hence no flux and so considered an average impulse 2p multiplied by an overall internal time in the system, namely hbar/E. We found that this scenario balanced with -dVd/x matches the quantum oscillator ground state. In this note we ask: What about other potential V(x) situations? Quantum mechanics seems to be a mixture of statistical mechanics and Newtonian dynamics i.e. the time-independent Schrodinger equation is, on average, an energy conservation equation for all energy levels i.e. KE(x)+V(x) = En where KE = {Sum over p a(p)pp/2m exp(kpx)}/{Sum over p a(p)exp(ipx)}. In (1) we took d/dx to bring out a force term -dV/dx and showed if is linked to <ppp/2m> and <pp/2m> , but analyzed these in terms of fluxes. The notion of flux makes sense using external time, but internally there is no time and so here we try to examine the situation in terms of various impulses.