For a Maxwell-Boltzmann gas (MB) it is known that T (temperature) is a statistical parameter related to the average energy. It appears in f(ei) = C exp(-ei/T) so T has the same units as ei, a single particle energy. If T increases, ei increases identically if ei/T is treated as constant. Thinking in terms of ei=pp/2m, this means the momentum spread (i.e. variance of the Gaussian increases) and one expects entropy to increase In (1) we argued that exp(-ei/T) for an MB gas and exp(ipx) for a quantum free particle (constant p), follow from the same statistical rule P(a1)P(a2)=P(a1+a2), but ei is a scalar and p a vector. Furthermore exp(ipx) is associated with spatial invariance and the generator of translations d/dx. In this note, we try to argue that a constant p may act as a statistical parameter just as T, although in this case hbar/p has the behaviour of a length i.e. the wavelength. If p increases (just like T is increased), the wavelength decreases meaning there are more peaks in a fixed length or more entropy, we argue. Thus there seems to be a statistical mechanical similarity to increasing T for an MB gas and to increasing p for a single free quantum particle. The association of x and p in exp(ipx) has implications, not only for the resolution of x space, but the increased entropy suggested by more peaks of cos(px). In addition, this means more peak places for impulses to occur. Thus a large p not only represents a large impulse, but is associated with more peak points for this impulse to be delivered and more entropy. In (2) we argued that the humps of cos(px) encode information about the impulse/force used to create p and which may be delivered if p hits an object. Thus there is a somewhat unusual relationship between p and x that one does not find in Newtonian mechanics, but has its roots in special relativity i.e. in the Lorentz invariant -Et+px. We suggest that just like exp(-V(x)/T) in an MB gas means that for a large force=-dV/dx, exp(-V/T) changes rapidly in x, a large p, associated with a large force should also imply that the humps in exp(ipx) rise (and fall) sharply, thus leading to sharp peaks. Thus sharp peaks are indicative of high internal force and increased entropy for a free particle exp(ipx), we argue.
For a Maxwell-Boltzmann gas (MB) it is known that T (temperature) is a statistical parameter related to the average energy. It appears in f(ei) = C exp(-ei/T) so T has the same units as ei, a single particle energy. If T increases, ei increases identically if ei/T is treated as constant. Thinking in terms of ei=pp/2m, this means the momentum spread (i.e. variance of the Gaussian increases) and one expects entropy to increase In (1) we argued that exp(-ei/T) for an MB gas and exp(ipx) for a quantum free particle (constant p), follow from the same statistical rule P(a1)P(a2)=P(a1+a2), but ei is a scalar and p a vector. Furthermore exp(ipx) is associated with spatial invariance and the generator of translations d/dx. In this note, we try to argue that a constant p may act as a statistical parameter just as T, although in this case hbar/p has the behaviour of a length i.e. the wavelength. If p increases (just like T is increased), the wavelength decreases meaning there are more peaks in a fixed length or more entropy, we argue. Thus there seems to be a statistical mechanical similarity to increasing T for an MB gas and to increasing p for a single free quantum particle. The association of x and p in exp(ipx) has implications, not only for the resolution of x space, but the increased entropy suggested by more peaks of cos(px). In addition, this means more peak places for impulses to occur. Thus a large p not only represents a large impulse, but is associated with more peak points for this impulse to be delivered and more entropy. In (2) we argued that the humps of cos(px) encode information about the impulse/force used to create p and which may be delivered if p hits an object. Thus there is a somewhat unusual relationship between p and x that one does not find in Newtonian mechanics, but has its roots in special relativity i.e. in the Lorentz invariant -Et+px. We suggest that just like exp(-V(x)/T) in an MB gas means that for a large force=-dV/dx, exp(-V/T) changes rapidly in x, a large p, associated with a large force should also imply that the humps in exp(ipx) rise (and fall) sharply, thus leading to sharp peaks. Thus sharp peaks are indicative of high internal force and increased entropy for a free particle exp(ipx), we argue.
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.
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.
A Maxwell-Boltzmann gas with no external potential V(x) has a constant spatial density. The value of this spatial density alone does not yield the pressure because this also depends on T through the ideal gas law Pressure = density(x) RT, where T is temperature (associated with a conserved average energy), but also the average single particle pressure. Taking d/density(x) of Pressure yields the single particle average pressure value RT, yet at the same time single particle pressure is a stochastic value evaluated as the average of constant v p where p is momentum and v velocity. The quadratic form of p arises, we argue, because one has more than one particle, thus v distinguishes between the particles and relative speed becomes important. In the case of a single particle (as in a single free quantum particle) moving at constant p (speed as well) there is again a constant spatial density as in the MB gas with no V(x). Given the value of the constant density 1/L (L=length), one has no information about momentum p. In the case of a single particle, impulse and not pressure is the important quantity, we argue, because there is no need for a v as there is no relative motion. One only needs to know impulse which is proportional to p. If one speculates that a free quantum particle may follow the model of pressure in an MB gas, one would require a function F(p,x) such that d/d something F = p where p is the single particle impulse. In the MB case, “something” is density, but this is not appropriate for a single particle as density consists of many particles. Without changing T, one may change pressure in an MB gas by changing density i.e. changing the x value of particles. For a quantum particle, we suggest that the conjugate variable to p is x. One may change x of the single particle. Furthermore an association between d/dx and p exists in Fermat’s Least Time Principle. T Thus we suggest d/dx F(p,x) = p where F is unknown both in form and meaning. Pressure, however, is a statistical quantity so F should also be a statistical quantity i.e. one could have d/dx P(p,x) / P(p/x) = constant p where P(p,x) is a kind of unnormalized probability. This suggests, however, that P(p,x) is an eigenfunction of d/dx and represents a kind of unnormalized probability. This, we argue, leads to the notion of exp(ipx) because one does not want a monotonically growing or decreasing probability in x for a constant p. The “constant” is then i. Thus, we argue that if one mimics the pressure behaviour of a MB gas, and applies its properties to a single particle moving at constant speed, one obtains an unusual unnormalized kind of probability exp(ipx) i.e. the quantum free particle wavefunction. At the same time, a constant spatial probability or density 1/L exists. The two are linked in that exp(ipx) exp(-ipx) =1. The overall idea is that the stochastic force i.e. the entropic pressure in an MB gas has an analogy even for a single particle with constant momentum.
Classical statistical equilibrium, for a Maxwell-Boltzmann gas for example, is based on localizing the gas to a certain volume and having no average changes in time. In other words, maximization of entropy subject to the average energy constraint sum over i ei P(ei) is linked with these two ideas. Ei (kinetic energy for no V(x)) is a conserved quantity in reactions and overall energy represents a quantity which does not change in time consistent with a center-of-mass which does not change and average values which do not change in time. This approach is associated with a specific kind of statistics, namely that of the constituents of the system. As a result, one obtains the Maxwell-Boltzmann distribution exp(-ei/T) where ei is an extensive property of a particle (or subset of the system) and T an intensive property linked with the whole system. Entire systems, however, may interact. For example, a N particle system with T1 may achieve equilibrium with a N system with T2 yielding a new average (T1+T2)/2. The probabilities exp(-ei/T), however, are for subcomponents of a system. One could argue that momentum is an additive (vector) quantity which is conserved in collisions, but the average momentum of the system is 0 which does not allow one to describe increases in average p and -p which may be described using energy, i.e. through T temperature. This begs the question whether momentum may be associated with maximization of entropy subject to a constraint. In (1), we argued that there were flow probabilities in quantum mechanics linked with exp(-iEt) and exp(ipx), which may be obtained formally through maximization of entropy. Here we argue that while an MB gas describes a localized volume with average values not changing in time, x and t state values change, even if p and pp/2m remain constant for a free particle and it may be possible to have a maximum entropy situation to describe the combination of px and -Et. In other words, we suggest localized versus nonlocalized equilibria. The latter represents a quantum free particle, but one may then try to localize such a particle in a potential. What happens in such a case? One does not switch to a localized maximization of entropy as in the MB gas case, because it is possible to have the potential leave the flow unbroken as argued in (1). In other words, the nonlocalized free particle equilibrium is akin to a particle bouncing back and forth in a box, which in classical mechanics is not considered a statistical process. Here we suggest that the underlying picture is statistical meaning that the classical mechanics approach seems to be an approximation. We consider both a quantum particle trapped in an infinite potential well and the ground state of an oscillator to examine ideas of maximizing entropy beyond exp(ipx). A key common feature of both types of equilibrium is the product rule P(ei)P(ej) = P(ei+ej) or exp(ip1x)exp(ip2x) = exp(i (p1+p2)x). We consider this feature as well.
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 a previous note (1), we argued that the classical statistical mechanics of a Maxwell-Boltzman gas is linked (in a canonical distribution sense) to an additive scalar( ei = kinetic energy of a particle) and leads to P(ei)= C exp(-ei/T) if one uses information loss considerations i.e P(a1)P(a2)=P(a3) where a is the additive scalar. We further argued that a quantum particle also satisfies this information loss equation, but with a being a vector such as momentum. This leads to exp(ipx) as a probability, although an unusual one. In (2) we argued that a classical equilibrium distribution based on an additive scalar which is linked by a conservation law in space i.e. a(at x1) + B(x1) = a(at x2) + B(x2) necessarily introduces a probability factor of the form exp(-iB(x)/T), but leaves the exp(-ai/T) distribution unchanged. We suggested this be interpreted as a spatial density linked to acceleration of particles. In this note, we argue that if both the quantum exp(ipx) and the classical MB distribution both follow from the same P(a1)P(a2)=P(a3) condition, then the notion of density and acceleration should also be found in quantum mechanics, even for exp(ipx) which represents constant velocity.