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.
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 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.
In a previous note (1) we argued that a quantum free particle, represented by exp(ipx), carries the information of the force that created it and which it may also impart when striking a target in the distributions cos(px), sin(px) of its wavefunction exp(ipx). In this note, we try to quantify this argument by first examining a second collective force in a quantum bound state other than -dV(x)/dx, namely KE(x) (-dW/dx / W) where W is the wavefunction. We argue this force should be present in a particle in a box with infinite walls and ultimately in a quantum free particle.Thus -id//dx, related to the translational operator, is linked to momentum and its conservation, but also ultimately to impulse carried by such a momentum, namely p. The force is found from pp/2m p where pp/2m is the frequency. Thus p is enough information for both momentum and force.. A high resolution (i.e. small wavelength) of a high momentum particle is directly linked to the high force it may impart when striking a target. Alternatively, the notion that an impulse Integral F dt occurs at a point x, as in two-body scattering in a Maxwell-Boltzmann gas, is only achieved in the high p limit, but an MB case is considered to be a high energy, high p limit i.e. a classical situation.
In previous notes, we presented a simple argument for obtaining the quantum free particle wavefunction exp(-iEt+ipx). We suggested that there are situations for which one cannot predict the outcome of an elastic collision and that given initial e1,e2 (energies) and p1,p2 (momentum vectors in 1-direction for simplicity), any ei,ej and pi,pj which conserve energy and momentum have the same product probability. Even in Newtonian elastic collisions of free particles, one sees calculations which allow any outcome which conserves momentum and energy. If the probability has the same real value for any free particle as no variable is being distributed and one wishes to impose Lorentz invariance, then exp(-iEt) and exp(ipx) represent the probabilities. This automatically gives rise to units in time and space, hbar/E and hbar/p (the hbar comes from considerations linked to a photon). One may ask, however: What is the real physics behind dx and dt? We argued in (1) that this physics is associated with force, in particular an intrinsic force. We considered a particle with rest mass mo and argued that this rest mass represents intrinsic force as it is proportional to Newton’s inertia and the force needed to set it in motion. If viewed from a moving frame, (0, mocc) maps to (E,p) with E containing the intrinsic inertial force as well as enhancements due to motion and p containing the intrinsic mo as well as the velocity vector. For a particle with rest mass, we suggest that a key idea is this notion of intrinsic force. We argued that one may consider the Lorentz invariant -Et+px = -mocc t1 and introduce dx=hbar/E, dt=hbar/p, dt1=hbar/(mocc) and dx1-> infinite as p->0. This leads to -E dt + p dx =0 or -E/dx + p /dt = 0. We then argued in (1) that p/dt is like an average force and from this we introduced exp(-iEt+ipx). Thus, dt and dx are linked with the notion of an average force. We suggest that this notion of intrinsic force is the reason behind linearizing -EE + p dot p = -momo cccc ((1)). -Edt+pdx=0 is too simple to show the geometry linked with rotational invariance etc in ((1)), yet at the same time ((1)) is not linear in the intrinsic force mo. This is the physical reason why ((1)) should be linearized via Dirac’s process, giving rise to spin ½. As a result, we postulate that a particle has an intrinsic force, in this case rest mass, which is directly linked with the probability exp(-iEt+ipx) and that one should be able to write an equation containing operators and the intrinsic force (in linear form) which also shows geometry. It is this equation which gives rise to spin. At this point, this is a postulate, i.e. speculation. It would be good if one could find an example in which these ideas arise naturally, i.e. a case in which one did not have to linearize an equation like ((1)) or make special arguments in order to obtain exp(-iEt+ipx). In that case, an equation linear in the intrinsic force would exist a priori and a direct link to exp(-iEt+ipx) would also naturally exist. We suggest that such an example does exist, namely for a photon. One may use the special relativistic equations developed for a particle with rest mass and set mo=0 to obtain E=[p] c for a photon. This is a fine equation, but unlike the rest mass case mo (proportional to the intrinsic force), there is no rest mass and no intrinsic force present for the photon. Certainly p (momentum) is linked with a force as both a particle with rest mass and a photon impart an impulse on a target, but momentum is not an intrinsic force, it is a consequential one. In the case of a rest mass, it arises because the rest mass (the intrinsic force) is seen to be moving. For a photon, one would then expect the existence of an intrinsic force which gives rise to momentum. We suggest that the electric field (together with the magnetic) represents this intrinsic force and that Maxwell’s equations allow one to directly write a linear equation in El (electric field) (no linearization required) which shows the geometry in the system. One may also explicitly show that El behaves as exp(-iEt+ipx) if one extends cos(-Et+ipx) to a more general complex number. In other words, the postulates made for the particle with rest mass mo seem to naturally arise in the case of the photon. This, we argue, is why the procedures for finding spin for a spin ½ particle with rest mass mo and a photon are so different even though both a photon and rest mass particle satisfy special relativity and are associated with exp(-iEt+ipx). It is the difference in the intrinsic force which is the key to finding spin, we speculate.
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.
Newtonian mechanics often employs a potential V(x) such that .5m v(x)v(x) + V(x) =E ((1)) where v(x) is velocity. Mathematically V(x) is defined at each x, but in reality there should be a tiny cutoff i.e. finite dx. Newtonian mechanics makes use of the idea of small dx lengths which tend to 0 mathematically. To first order v(x) is constant within a dx range, but the overall mathematics is continuous instead of discrete. Thus there is a continuous equation ((1)) for conservation of energy. A photon may be described by an eclectic field varying as cos(-Et+px). In terms of wavelengths exp(ipx) is often used as a mathematical convenience and noncontinous behaviour is not an issue. For example in refraction, momentum = hbar/wavelength abruptly changes and one straight line path is joined to another with a different slope. exp(ipx) is associated with repeatable wavelength patterns with a single wavelength representing a spatial distribution/ (kind of probability) region. A particle with rest mass may also be described by exp(ipx) which suggests a characteristic length hbar/p. Within this length (i.e at each x point which exist within hbar/p) the particle has p and pp/2m. Thus for a square well potential there is conservation of energy at each x point even though there is a wavelength in the problem. This is fine for a constant momentum, but what about a potential V(x) which implies p must change at each dx region? exp(ipx) is an eigenfunction of -id/dx which is linked to spatial invariance. It manifests itself physically by interference (e.g. two slits) which again involves non-smooth motion (i.e. one compares rays with completely different slopes). The characteristic length hbar/p is fixed and associated with a single p, yet momentum is supposed to change from one x point to the next. hbar/p does not fit within a single dx and this is a problem. Quantum mechanics remedies this issue by allowing for a superposition of exp(ipx)s i.e. W(x) = wavefunction = Sum over p a(p) exp(ipx) such that energy conservation is forced at each x. Thus one combines the idea of wavelength (which is physical) with energy conservation which is also physical. A consequence of this, however, is that -id/W/dx / W leads to a strange average momentum which is not the square root of -1/2m d/dx dW/dx/ W as well as discrete energies. A mathematical benefit is that one has a continuous differential equation, namely the time-independent Schrodinger equation. The question we ask is: What happens in the so-called classical limit? In the nonrelativstic case this leads to Newtonian mechanics in which there is no wavelength. How does the wavelength disappear so to speak? We suggest (as has been suggested often in the literature and textbooks) that the wavelengths in W(x) may become extremely tiny so as to not be detected easily in classical length scales. Mathematically the time-independent Schrodinger equation holds and there is a smooth W(x) at all points, but we suggest that this smoothness may not be necessary because it is not found in refracting light. If the wavelength is tiny compared to small dx regions in the classical limit, then many wavelengths of a given p fit into dx. Thus the idea of spatial invariance approximately holds within dx and there is no issue of the wavelength spilling out of the dx region as it does when it is comparable to the size of the system as in quantum bound problems. Thus within dx a single p with many (or at least several) wavelengths holds and there is a single p. The kinetic energy is pp/2m and the potential V(x) where x is in the center of dx. One may think of p changing abruptly (nonsmoothly) at the boundary of one dx with another such that conservation of energy holds. Thus one has p wavelengths which do not spill out of dx and give an approximate sense of spatial invariance together with energy conservation. The only loss is that of a smooth wavefunction, but it is not clear that such smoothness is a requirement. V(x) is physically associated with a limiting dx length (at which the concept of V(x) does not hold, but rather there are stochastic hits) and if one expands this length i.e. chooses a much bigger dx in keeping with classical scales, it seems that there should not be a problem. It as if V(x) delivers an impulse hit at each dx junction (like a series of different media in refraction).. This approach leads to the decoherence of wavelengths in the classical limit.
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.