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.
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.
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.
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 previous notes, we argued that one may introduce a probability into Newtonian elastic scattering if one assumes that for an incident (e1,e2) energies and (p1,p2) momenta vectors, any (ei,ej), (pi,pj) which satisfy conservation of momentum and energy have equal probability. We then found that for a single particle, the relevant probability is exp(-iEt+ip dot r). This is a unit modulus complex number which is Lorentz invariant, and exp(i p dot r) is time reversal invariant, i.e. p→-p and r→-r yield the same value. The question then becomes: How does one use exp(-iEt+ipx) or for time-independent problems, exp(ipx)? This is not a usual probability linked to quantity, e.g. N P(reflect) = the number of particles which reflect. exp(ipx) is a probability to have a certain momentum, but there is no reason it cannot carry a weight, i.e. a math factor. The question is: What does such a factor mean physically? Secondly, exp(ipx) depends on x (space), but exp(ipx) is not the probability to be at x, but it is somewhat related in that for any x and a given p, the modulus has the same value, suggesting equal probability to be at any x. This mapping, however, really only holds for a single exp(ipx). It seems that one can say two things about exp(ipx). First, if it is a probability for a momentum p, then one should be able to write p exp(ipx) or function(p)exp(ipx). Secondly, one should be able to write AND (multiply) and OR (add) statements. The idea of an OR or addition of exp(ipx)s is important, because one may be able to write an equation for constant kinetic energy: -1/2m grad dot grad W(x) = numberW. The general solution should be a W(x) which contains a sum of all exp(ipx) type solutions which have this kinetic energy. (In the relativistic case, one may consider d/dt d/dt W - cc grad dot grad W = -momocccc W.) The question we consider in this note is: Given that one may write W and grad W, what can only really do with these? They do not directly represent physical observables. Furthermore, if one wishes to impose continuity on W and gradW, which is reasonable mathematically, one should have a physical argument to justify it, and we suggest that this is a little complicated. Ultimately, we try to argue that one may consider continuity of W as well grad W in certain cases based on the association with a classical quantity probability equation. We also suggest one may relate W* grad W which may be generalized to W* grad W + (grad W*) W, as discussed in Part 3 and provides a link to classical probabilities.
Quantum free particles are represented by exp(ipx) which we argue is a dynamical probability which maps into a static probability exp(-ipx)exp(ipx) i.e. the usual spatial density. Shannon’s entropy = - Sum over i P(i) ln(P(i)) is often associated with the static probability P(x)=W*(x)W(x) (1), but in (2) we argue that one may introduce a different probability C1 Real(W(x)). In this note we consider both in different examples. In particular we argue that the two different probability definitions are ultimately linked with momentum conservation. The form W*(x)W(x) introduces local nonconstant spatial densities if there are mixed exp(ip1x) and exp(ip2x)’s within W(x). If there are not, this form W*(x)W(x) does not distinguish space i.e. exp(ipx)exp(ipx+ib) is still a constant. Nevertheless interference may still occur which may be linked with entropy. Thus we first examine the case of exp(ipx) + exp(ipx + ib) where b is a constant phase shift. W*(x)W(x)=constant which does not lead to a structure different from exp(-ipx)exp(ipx), yet W(x)=C2{exp(ipx) + exp(ipx+ib)} interfere. On the other hand |cos(px)| and | cos(px)(1+cos(b)) - sin(px)sin(b) | are not the same distributions (one is a parity eigenstate, the other is not) and so we argue there is a different local entropy density. Thus introducing extra information i.e. b changes local density and entropy density if one adjusts the definition of probability used. We next consider W(x)= exp(ip1x)+exp(ip2x). One may also use the probability C3 Real|W(x)|, but here we are interested in an association with conservation of momentum. For a given exp(ipx), exp(-ipx)exp(ipx)=1. We argue that this represents a steady state picture i.e. a kind of equilibrium in which all x points are treated the same. W*(x)W(x) breaks this equilibrium scenario by introducing a term 2 cos( (p1-p2)x ). Thus both local spatial density and local entropy (based on W*W) are different and do not represent the “equilibrium” scenario. We further argue that momentum conservation may be linked with this equilibrium type of density and entropy density at least in the case of the transition amplitude <p1 V(x) p2>. V(x)exp(ipx) yields a sum of exp(ipx)’s, but only exp(ip1x) survives the integral over dx. The others create spatial densities which are not constant and so represent local entropy densities which do not treat all x points the same i.e .is not an equilibrium one. Thus we try to use Shannon’s entropy with two different probability definitions to analyse different quantum interference scenarios and argue that momentum conservation (and possibly others) may be linked with local entropy density in some situations.
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.
In classical statistical mechanics, e.g. a Maxwell-Boltzmann (MB) gas, one tries to find the probability distribution of a particular quantity such as particle energy pp/2m. Thus there is a distribution in a quantity, making it the stochastic variable. This quantity, however, is subject to a conservation law i.e. elastic two particle scattering. In a previous note (1), we considered the case of reflection/refraction and called momentum the stochastic variable and noted that it too was linked with conservation. In fact, we argued that both the MB gas and reflection/refraction followed the same stochastic law: P(variable1) P(variable 2) = P(variable 1 + variable 2) whether variable 1 is a scalar or vector. In (2), however, we noted that exp(ipx), a free particle wavefunction, treats p as a parameter with x being the “probability” variable. This seems a little unusual because p, which is normally a variable in an MB gas and even in a quantum bound state i.e. a*(p)a(p) where W(x)=wavefunction = Sum over p a(p) exp(ipx) takes on the role of a parameter i.e. p=hbar/wavelength. Why this duality? Why is p a parameter for “x” of the probability distribution and yet at the same time a variable associated with conservation of momentum? In this note, we argue that unlike the distribution of a defining property of a particle such as momentum or energy, one may have a distribution associated with probability itself. This we argue is the case for a reflection/refraction problem which applies to both light and a particle. In the case of light, there is an interface of two media with different indices of refraction and in the case of a particle an interface with two constant potentials. In both cases, total energy of the photon/particle is constant, but there are two choices or probabilities, reflection or refraction, but they are linked with different times. A single photon or particle can only undergo one at a time. One knows exactly the two values of momentum, but the probability of each is unknown. Thus unlike an MB gas where energy/momentum is being redistributed because the two particles of the outcome exist at the same time, here a probability of 1 must somehow be distributed to the two events.. Given that the incident momentum is p, the reflected -p and the refracted p2, only considering a specific time event i.e. reflection or refraction, does not conserve momentum. In the MB case, a single time event i.e. scattering conserves both momentum and energy. Thus it seems that one does not deal with events in time. There is a distribution of probability in such a way as to ensure conservation of momentum, but does this probability already exist or get created at the interface point? We argued in (3) that if the scenario were deterministic one would have to follow it in time and treat each event separately. Rather we suggest that a distribution of probabilities allows for conservation of momentum, but this means that all three momenta exist at the interface x point as one is not considering an event at a precise point in time. This then allows for each momentum to have its own x distribution based on momentum p so that one may impose conservation of momentum and not have a deterministic time associated with the interface point.. This last idea of conservation of momentum appearing is not discussed in (3) and allows for p to behave like a parameter for x while at the same time being a variable which is conserved. In short, in an MB gas one distributes energy between two colliding particles in order to conserve total energy while for a quantum particle or photon which reflects/refracts one distributes probability to the two scenarios (reflection or refraction) such that momentum is conserved. Thus the nature of what is “distributed” is very different.