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.)
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.)
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 and classical statistical mechanics/thermodynamics are both statistical theories, but have key differences. We argue that one such key difference is nonlocality in the quantum world i.e a wavelength proportional to 1/p and a probability of exp(ipx) associated with p versus locality in the classical case. Quantum nonlocality tends to lead to spatial densities with humps even for a particle with a fixed energy in an infinite potential box. Recently it has been suggested in (1) that one may think of a quantum particle in a box with infinite walls in terms of thermodynamical properties e.g. pressure which is taken to be constant throughout the box. From this assumption, one finds that quantum entropy is ln[ Average(nxnx + nyny+nznz)]. For a single quantum state, this seems to imply that entropy is ln[nxnx+nyny+nznz] ((1a)). In a previous note (2), we suggested that quantum entropy for a wavefunction (say an eigenstate) is Integral dx W*(x)W(x) ln[W*W] + Integral dp a*(p)a(p) ln[a*(p)a(p)] ((1b)). We found this is independent of L for a particle in a box with infinite walls so that displacing the wall adiabatically should mean that dS/dL = 0 which it does as L cancels from ((1)). We argue in this note that quantum pressure should be linked to the density humps for a sin(pave x)sin(pave x) wavefunction even though pave is constant at all x points. Thus it seems that one would not consider a constant pressure for sin(pave x)=wavefunction. Furthermore ((1b)) does not yield the same entropy as ((1a)). It is noted in (1) that ((1a)) is an exact expression for entropy and that the Maxwell-Boltzmann distribution entropy approximates it. In this note, we argue that a time reversal balanced scattering reaction leads to the Fermi-Dirac and Bose-Einstein distributions and these both lead to the Maxwell-Boltzmann distribution under certain conditions. (The Maxwell-Boltzmann distribution may also be obtained from time reversal balanced scattering with looser restrictions.)
Historically, thermodynamics was formulated for macroscopic systems followed by the development of statistical mechanics which provided microscopic formulations for quantities such as entropy. Entropy is traditionally linked to the ln of the number states available or to the Shannon’s form -k Sum over i P(ei) ln(P(ei)) where P(ei) is the probability for the quantity ei. In addition a variable T (linked to average kinetic energy) appears in both theories. Later the Schrodinger equation was developed to describe bound electron states in an atom etc. These states were characterized by a math object W(x) called the wavefunction and a set of discrete energy levels was calculated which matched data of emitted photons. In a previous note (1) we argued that the average energy En for a quantum bound state plays the same role of characterizing equilibrium as T in classical thermodynamics/statistical mechanics. We further argued that En is linked to parameters in the system. For example, for a particle in a box with infinite potential walls, En is linked to L as well as to n the energy level. One may change En by either changing n or L separately (or both). Holding L constant and changing n leads to a certain type of energy change. We argue this is similar to holding the volume of a gas constant and increasing T. Alternatively, one may leave n unchanged (a quantum adiabatic process) and change L i.e. a mechanical change or work. We argue that one may already consider the concept of the first law of thermodynamics at the quantum bound state level i.e. dE = dQ - dWork where Work is linked to a change in a physical parameter, such as L the length of the box, whereas dQ is linked to energy changes not involving a physical parameter e.g. a change in n the energy level. The Schrodinger equation yields solutions for W(x)=Sum over p a(p)exp(ipx) the wavefunction as well as En’s. We have argued in previous notes that W(x) is part of a conditional probability P(p/x)= a(p)exp(ipx)/W(x) and that En is constant at each point x just as T is constant at each x for classical statistical equilibrium (1). Thus a kind of statistical mechanics exists at the wavefunction level we argue. This statistical mechanics makes use of a free particle wavefunction exp(ipx) and weights a(p), but does not contain the usual P(x), P(p) picture of the classical world. We have shown in (2) that one may obtain P(x)=W*(x)W(x) and P(p)=a*(p)a(p) from the conditional probability. The essential physics we argue, however, is at the level of the wavefunction together with its own unique statistical scenario of exp(ipx)’s interacting with exp(ikx)’s from V(x)=Sum over Vkexp(ikx) (see (1)). Given that P(x) and P(p) exist for a quantum bound state, it may be convenient to use them in formulas because they are linked to a classical statistical picture. We argue that one may use P(x) and P(p) to construct an object S such that T dS does not change when one changes L, the size of box with infinite potential walls. We argue that such an object follows from properties of the Schrodinger equation together with various assumptions. We arrive at S=Sp+Sx where Sp=-k Integral dp a(p)a(p) ln{|a(p)a(p)}| and Sx= -k Integral dx W(x)W(x) ln(W(x)W(x)) for a given energy level. We do not use arguments related to taking ln of the number of states present or Shannon’s entropy. Rather we look for a solution which does not depend on L and is additive. We note, however, that for both W(x)W(x) and a(p)a(p) the solution found is mathematically consistent with the entropy form found by considering independent probabilities. This, however, applies to P(x) and P(p), but we argue that the physical situation is based on W(x) and a(p) as solutions of the Schrodinger equation. Nevertheless P(x) and P(p) are physical observables and not simply math constructs and so may think of them in a somewhat classical manner in terms of probability. Thus we argue that thermodynamics is already present at the level of a quantum bound state and that one may obtain Shannon’s entropy at this level using P(x) and P(p), but without considering the overall probability of independent probabilities as in information theory or numbers of states. Rather obtaining a function which is additive in Sp and Sx and eliminates L is the main consideration. We also argue that S=Sp+Sx carries over into mixed quantum state wavefunctions even though it is usually ignored. Thus we try to argue that it is the bound quantum state which gives rise to the notion of thermodynamics and entropy.
Classical statistical mechanics is often derived using counting methods applied to all possible configurations a system may take subject to a total volume and a total energy constraint. Given that there are numerous configurations, there should also be an average of these configurations described by f(e) for the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) distributions. In previous notes we have argued that one may apply a dynamical constraint to the average configuration. In particular, we argued that there should be a time reversal balance for two body elastic collisions. This immediately yields the various distributions with only consideration given to the average state and no counting techniques used. The counting of configurations, however, emphasizes that statistical mechanical systems contain “fluctuations” and that a specific dynamic constraint (i.e. time reversal two body scattering) applies only to the average. Quantum bound states, we argue, are very similar. They represent a statistical system which has many fluctuations (i.e. a particle may be knocked out with an momentum p with probability a*(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx)). There is, however, also a dynamical constraint which applies to the average, namely KE(x)+V(x)=E which is a classical mechanical constraint. Here KE(x)= [-1/2m d/dx dW/dx /W] ((1)). If fluctuations disappear in the high energy scenario, one may assume there is an almost fixed momentum p at each x. ((1)) still applies and there is still a wavefunction which is often in the form of crests and troughs. If an almost unique p exists at each x, one expects W(x) to vary in height at peak values. This may be verified from actual W(x) solutions. Given that the constraint on the statistical average is KE(x)+V(x)=E, one would expect that as p becomes an almost unique value at each x, that the quantum expectation Integral dx WW KE(x) with KE(x) given by ((1)) should also approach a classical value in other words Integral dx WW —> integral dt. One does not need to impose the correspondence principle to arrive at this result, one only needs to consider the case in which fluctuations in a quantum system disappear allowing the average p at each x to consist of almost a single p value.
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.