We argue that classical statistical mechanics (say of an ideal gas) is based on a single equilibration scheme characterized by a temperature T (even in the presence of a potential V(x)). This single scheme is based on the maximization of entropy (subject to an average energy constraint) in order to determine probabilities for p,v and x. Quantum mechanics, on the other hand, is based on two equilibration schemes, we argue. The first is due to a built-in probability of exp(ipx) for a free particle. In this case, a potential V(x) interacts with momentum (impulse hits) to create a potential energy. No velocity appears in the creation of potential energy, but does appear in the creation of average kinetic energy. Like with a classical mechanical bound state, energy becomes a system variable. This is the first equilibration. We argue that it should have its own entropy. The probability form exp(ipx) together with conservation of energy and boundary conditions fix allowable single particle energy values (which are discrete) unlike in the classical statistical mechanical case. The second equilibration occurs if there exists a mechanism for moving from one discrete E to another, e.g. photon emission/absorption. This equilibration is based on maximization of entropy (a second entropy), but the effects of the first equilibration remain because it is this first scheme which controls allowable E values. We also argue that exp(-i Et) from the first equilibration scheme may affect the usual entropy based on exp(-E/T) used in the second scheme.
We argue that classical statistical mechanics (say of an ideal gas) is based on a single equilibration scheme characterized by a temperature T (even in the presence of a potential V(x)). This single scheme is based on the maximization of entropy (subject to an average energy constraint) in order to determine probabilities for p,v and x. Quantum mechanics, on the other hand, is based on two equilibration schemes, we argue. The first is due to a built-in probability of exp(ipx) for a free particle. In this case, a potential V(x) interacts with momentum (impulse hits) to create a potential energy. No velocity appears in the creation of potential energy, but does appear in the creation of average kinetic energy. Like with a classical mechanical bound state, energy becomes a system variable. This is the first equilibration. We argue that it should have its own entropy. The probability form exp(ipx) together with conservation of energy and boundary conditions fix allowable single particle energy values (which are discrete) unlike in the classical statistical mechanical case. The second equilibration occurs if there exists a mechanism for moving from one discrete E to another, e.g. photon emission/absorption. This equilibration is based on maximization of entropy (a second entropy), but the effects of the first equilibration remain because it is this first scheme which controls allowable E values. We also argue that exp(-i Et) from the first equilibration scheme may affect the usual entropy based on exp(-E/T) used in the second scheme.
Both quantum bound state equilibrium and classical statistical mechanical equilibrium are characterized by d/dt partial density(x,t)=0. For a quantum particle, W(x,t)=exp(-iEt)W(x) and d(x,t)=W*W. Density is linked to fluid flow and arguments of pressure balance such that the overall average fluid flow is zero are made. This pressure-force balance is common in classical statistical mechanics and used to calculate the variation of density with x i.e. obtain a differential equation with d d(x,t) /dt partial. A similar balance equation is used in (1) as the authors try to derive the Schrodinger equation using a fluid dynamical approach. In this note, we try to argue that one may think of the two equilibria in terms of an energy quantity which remains constant throughout space. This energy quantity depends directly on the type of interactions in the system at any point x. For a classical ideal gas, balanced two body time reversal scattering represents the interaction. V(x), a potential, does not change this fundamental manner of interacting at any given point, although it does affect density. As a result the energy related constant at each x is T, related to average kinetic energy. We argue that this already points to the dominant type of interaction creating the statistical interaction i.e. one that depends only on p in interactions which conserve kinetic energy. This “happens” to be equivalent to maximizing Shannon’s entropy, but we argue this is particular to this scenario in the sense that not all interacting systems will have this equivalence. In other words, we argue that one must be careful with regards to the maximization of entropy as a fundamental rule. In particular, we consider the single particle bound quantum state. The interaction exists between the particle and the potential V(x), but one may consider a distribution of free particle momentum probabilities i.e W(x)=wavefunction = Sum over p a(p)exp(ipx). The form exp(ipx) comes from separate arguments as is discussed in the note, but the point is that there are no two body scattering reactions occurring at a given x, neither is there acceleration from on x to another (except for a vrms mathematical quantity). Nevertheless the “free particle” exp(ipx) interacts at each x with V(x). As a result of this interaction, one has free particles exp(ipx) which are no longer associated with pp/2m, but rather with an Eave which is a constant for every p. In other words the identity of exp(ipx) is unusual because it is linked to exp(ipx) and pp/2m, but also to Vk1 a(p-k1), Vk2 a(p-k2) etc where V(x)=Sum over k Vk exp(ikx). Thus each x point is associated with a constant Eave and this defines the equilibrium. (It is also equivalent to conservation of energy at each x.) In such a case, one does not maximize entropy to find a(p) values. In the two cases, the common point seems to be considering the interactions at each x and finding a constant “energy” type quantity which does not vary with x. This we argue appears to define equilibrium for both the classical statistical and quantum bound states. Instead of considering density which does not change in time at each x, there is constant energy which not only does not change in time, but has a constant value at each x which seems to be an alternative rule.
Quantum mechanics is often compared with classical mechanics because both may deal with a single particle subject to a potential. Classical mechanics is said to be deterministic describing x(t) and its derivatives i.e showing how a particle moves and accelerates while bound state quantum mechanics produces a real wavefunction W(x) such that W(x)W(x) yields a spatial density profile. The comparison is carried further with the correspondence principle which states that at high energy levels quantum mechanics should mimic classical mechanics. Classical mechanics does not simply describe x(t) and its derivatives i.e. deterministic motion. It also includes two body scattering, force and torque balances for a stationary object etc. In the case of the classical statistical mechanics of an equilibrium ideal gas, the Maxwell-Boltzmann (MB) distribution in the absence of a potential is C1 exp(- pp/(2mT) ). If a potential V(x) is added, one may find P(x) two different ways using Newtonian ideas. The first is to consider the gas as a collective entity like a tiny macroscopic classical object at x. Then one may argue that equilibrium means the system is stationary and apply Newtonian statics i.e. a balance of pressures and -dV/dx to obtain exp(-V(x)/T) as is often done in textbooks. Alternatively, one may consider a single particle and apply Newtonian mechanics differently, namely use a conservation of energy equation: p1p1/2m + V(x1) = p2p2/2m + V(x2). Again one obtains exp(-V(x)/T). The spatial part of probability may be based on Newtonian mechanics indicating that it is not stochastic. The stochastic function is C1 exp(-pp/2mT). Interestingly, however, the two Newtonian approaches are different with respect to a single particle. A collective nonmoving set of particles acts as a stationary object, yet within it each particle seems to be “bouncing” off of an imaginary wall creating pressure. Thus one has an impulse free particle picture yet obtains the correct result exp(-V(x)/T). Alternatively one may think of a single particle accelerating which it does if it is not colliding with other particles. This is a very different picture of the single particle than of one bouncing back and forth hitting an imaginary wall. In fact, there is usually a wall holding the gas together and pressure and bouncing particles are the real picture there, but within the gas one would normally think of a single particle accelerating. Nevertheless there are two scenarios which yield the same result. The idea of pressure balance is more in keeping with an overall statistical picture of the gas. In other words, one may actually ignore the idea of following a single particle as it accelerates. (That is not to say that single particle acceleration does not exist - it does.) Quantum mechanics we argue is a statistical theory. In the case of a single bound particle one has a momentum distribution W(x)=wavefunction = Sum over p a(p)exp(ipx). One may apply conservation of average energy at each x to obtain the time-independent Schrodinger equation. In such a picture, like with the statistical mechanical gas, we argue that one may imagine impulse hits with the potential instead of considering acceleration. The quantity vrms(x) from KEclassical= .5m vrms(x)vrms(x) = -1/2m d/dx dW/dx / W does display acceleration, but this is a mathematical average. As energy levels become higher the correspondence principle is said to apply, but a classical particle exhibits acceleration. The quantum bound state does not, but like in the case of the statistical mechanical gas it may be that two scenarios lead to the same result. In other words, the quantum bound W(x) still has crests and troughs, but with low resolution one may only see an envelope roughly touching W(x)W(x) which matches a classical C2/v(x) where v(x) is classical velocity. Given that quantum mechanics is a statistical theory it may not be necessary to have it explicitly display acceleration.
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.
Classical statistical mechanics often utilizes the idea of maximizing entropy subject to a physical constraint For example, for a Maxwell-Boltzmann gas, this constraint is Sum over i ei p(ei) where ei=pp/2m (no V(x)). We have argued previously that there may be many possible constraints, i.e. an average of any power of ei could be a physical constraint. We stated that a conserved quantity associated with interactions present (creating equilibrium) should be used. Here we examine the constraint in terms of mechanical resistance i.e. pressure. We further consider the free particle quantum wavefunction exp(-iEt+ipx) in terms of resistances in time and space. We argue that the form exp(-ipx)exp(ipx) = 1 removes the stochasticity associated with motion due to the resistance (impulse) p Thus 1 or 1/L (L=length) seems to be a smoothed over picture associated with a constant flux and overlooks the inherent stochasticity associated with motion. This stochasticity is crucial when describing impulse hit reactions (n-slits, index of refraction interfaces, interactions with V(x)), but if a particle does not interact in space, then the W*(x)W(x) picture seems to be used to describe the v(average)=x/t scenario. The issue is that for a quantum bound state, there is also an rms velocity based on a wavefunction and one may create an average W(x)W(x) (W(x) is real for a bound state), but like exp(-ipx)exp(ipx), this bypasses some of the stochasticity in the problem, we argue. It only demonstrates stochasticity associated with differences in momentum.
Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)
Maximization of entropy subject to a constraint Sum over i ei P(ei) = Eave may be used to determine the Maxwell-Boltzmann factor exp(-ei/T), where T is temperature. In the case of a potential V(x), one may maximize -P(x) ln(P(x)) subject to the constraint TV(x)P(x). Average potential energy is not an a priori known value, but we have argued in the past that maximization of entropy subject to a constraint, followed by a derivative leads to a balance equation. What about a classical mechanical bound problem? This is usually considered nonstatistical . We suggest it may be a statistical problem, but one for which maxent applies to flow, in particular the quantum bound problem. We have argued before that for a free particle, one may maximize Shannon's entropy -P(t)ln(P(t)) subject to a periodic constraint i Et P(t) and -P(x)ln(P(x)) subject to i px P(x) to obtain exp(-iEt) and exp(-ipx). We argue these are associated with flows in time and space, and that there are problems in nature based on such flow and its interference. Classically, a bound system is a flow problem because a particle moves to the right and then left, between turning points. This means the particle is confined or localized,, but exp(ipx) is not. exp(ipx) is two dimensional to account for motion in a certain direction, but both cos(px) and sin(px) exist everywhere in space. A bound system should not have one directional motion,on average, thus either cos(px) or sin(px) applies, i.e. a certain parity. How does one account for a probability value in all of space? We argue that in the flow exp(ipx) picture, one must consider a bound problem in equilibrium with particles trying to enter or escape. By creating an ensemble W(x)=Sumo v over p a(p)exp(ipx) and having W(x) tend to 0 at x= +-/infinite one may create such a flow picture. At first one may consider any a(p) values, but we argue that either a(p)=-a(p) or a(p)=a(-p) for all p in this note, i.e. there is no mixing of parity. We try to show this. We then argue that V(x) does not impede this flow statistics, but only imposes conditions on the average kinetic energy at x. Thus Newtonian mechanics deals only with this restriction and not with the underlying average of maxent flow probabilities exp(ipx). A classical statistical gas is not a flow problem, but a balance problem with or without V(x)..Interparticle collisions change the kinetic energy of a particle and hence break its exp(-iet) flow probability. The stochasticity is then not in the flow, but in the flow-breaking collisions We suggest that this is linked to maximization of entropy subject to a constraint and deals with different probabilities than flow ones (exp(ipx) and exp(-iEt). A Rabi frequency problem on the other hand is a flow in time problem with the magnetic field not inhibiting this flow. Finally, there may be problems which at first appear to be a balance problem, but are really a flow one, like one dimensional photon reflection/refraction.