The Maxwell-Boltzmann MB distribution is often derived by maximizing the number of possible arrangements of particles with different energies subject to the constraint of a fixed number of particles and fixed energy. Using the canonical approximation that only average energy is conserved and using Stirling’s approximation for large number factorials leads to the Maxwell-Boltzmann distribution. The ln of total number of arrangements using Stirling’s approximation is linked to a math form called Shannon’s entropy i.e. - Sum over i f(ei) ln(f(ei)). Interestingly this same MB result may be obtained through reaction balance which makes no use of the idea of arrangements or entropy or maximization. It is concerned with including all possible physical reactions (for example two body reactions between all energy particles) subject to conservation of energy. In this sense it deals directly with the dynamics of the problem while the arrangement approach does not. The dynamical approach considers all possible reactions with notions of independence or lack of information and ensures the system remains unchanged. Why are they the same? We argue that it is because the “recipe” for the probability of two body reactions namely n(e1)n(e2) is of the same form as the leading factor of n(e1)! n(e2)!. In the case of quantum mechanics one may apply the dynamical approach to obtain the time-independent Schrodinger equation. In such a case one does not need to think in terms of “maximizing” any quantity although one considers all possible reactions with a potential V(x). a(p) factors in W(x) = wavefunction = Sum over p a(p)exp(ipx) are interrelated, but retain their identity after all possible interactions occur. This we argue is the idea for the eigenvalue equation. At the same time, Shannon’s entropy, which is directly linked to the notion of arrangements, is applied to a bound state and possibly linked to thermodynamics i.e. dE = TdS -PdV which may be applied to an adiabatic expansion. Shannon’s entropy is created from existing probabilities in this case which do not arise from explicitly maximizing entropy subject to constraints as the probabilities follow directly from the Schrodinger equation. One might ask why Shannon’s entropy has any relevance if it is the dynamics which determines the value of W(x) and a(p). It seems, however, that one may knock out a particle in a state p with probability P(p)= a*(p)a(p) or find a particle at x with probability P(x)=W*(x)W(x). These are classical type probabilities linked to the quantum dynamical probabilities a(p) and W(x). It seems Shannon’s entropy is only applied to P(x) and P(p) or a product P(x)P(p) which may give it physical meaning in a quantum bound state. We examine these ideas in this note.
The Maxwell-Boltzmann MB distribution is often derived by maximizing the number of possible arrangements of particles with different energies subject to the constraint of a fixed number of particles and fixed energy. Using the canonical approximation that only average energy is conserved and using Stirling’s approximation for large number factorials leads to the Maxwell-Boltzmann distribution. The ln of total number of arrangements using Stirling’s approximation is linked to a math form called Shannon’s entropy i.e. - Sum over i f(ei) ln(f(ei)). Interestingly this same MB result may be obtained through reaction balance which makes no use of the idea of arrangements or entropy or maximization. It is concerned with including all possible physical reactions (for example two body reactions between all energy particles) subject to conservation of energy. In this sense it deals directly with the dynamics of the problem while the arrangement approach does not. The dynamical approach considers all possible reactions with notions of independence or lack of information and ensures the system remains unchanged. Why are they the same? We argue that it is because the “recipe” for the probability of two body reactions namely n(e1)n(e2) is of the same form as the leading factor of n(e1)! n(e2)!. In the case of quantum mechanics one may apply the dynamical approach to obtain the time-independent Schrodinger equation. In such a case one does not need to think in terms of “maximizing” any quantity although one considers all possible reactions with a potential V(x). a(p) factors in W(x) = wavefunction = Sum over p a(p)exp(ipx) are interrelated, but retain their identity after all possible interactions occur. This we argue is the idea for the eigenvalue equation. At the same time, Shannon’s entropy, which is directly linked to the notion of arrangements, is applied to a bound state and possibly linked to thermodynamics i.e. dE = TdS -PdV which may be applied to an adiabatic expansion. Shannon’s entropy is created from existing probabilities in this case which do not arise from explicitly maximizing entropy subject to constraints as the probabilities follow directly from the Schrodinger equation. One might ask why Shannon’s entropy has any relevance if it is the dynamics which determines the value of W(x) and a(p). It seems, however, that one may knock out a particle in a state p with probability P(p)= a*(p)a(p) or find a particle at x with probability P(x)=W*(x)W(x). These are classical type probabilities linked to the quantum dynamical probabilities a(p) and W(x). It seems Shannon’s entropy is only applied to P(x) and P(p) or a product P(x)P(p) which may give it physical meaning in a quantum bound state. We examine these ideas in this note.
Historically statistical mechanics was developed for gasses for which there are very large numbers of molecules (i.e. on the order of 10 to 23rd power). As a result factorial schemes were developed to count the number of arrangements possible, the ln taken of this result and Stirling’s approximation for large N applied. The Shannon’s entropy form S/(-k) = Sum over i P(ei)ln(P(ei)) is consistent with this large N approach. Furthermore, large N statistical mechanics was linked to thermodynamics, in particular the first law dE=TdS - PdV. Surprisingly then, statistical mechanical results were applied to systems with small numbers of particles (e.g. a nucleus with about 100 nucleons etc). In Parts I and II we argued that the first law of thermodynamics describes two separate kinds of changes associated with energy changes. The second, work, is linked to the change of a mechanical parameter such as the volume of a box etc. The energy of a bound quantum particle in a box with an infinite potential is proportional to nn/LL. Thus we argued that a pure quantum bound state may also be associated with the first law of thermodynamics with changes in L with n, the energy level, held constant representing pure work. We further argued that for a bound state two classical like probabilities exists P(x)=W(x)W(x) and P(p)=a(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx) with P(x) being proportional to 1/L and P(p) to L. For a quantum adiabatic transformation (i.e. changing L with n staying constant) the term dS=0. If one wishes to create S from P(x) and P(p) and desires linearity i.e. S=Sp(n)+Sx, then Sp=Integral dp P(p)ln(P(p)) and Sx= Integral dx P(x) ln(P(x)) is a solution. If one next considers a quantum wavefunction for two levels W(x,t) = a(1)exp(-i E1 t) W1(x) + a(2) exp(-iE2 t) W2(x) with Eave = Sum over n en P(en) where Wn=a(n)a(n), one may again utilize the first law of thermodynamics. In Parts I and II we carried over the entropy form P(ei)ln(P(ei)). In this note we wish to argue that this form follows at the two energy level picture by considerations of a function which consists entirely of probabilities i.e. P(ei) and is linear in the sense that such a function for two identical two state systems should yield S1+S2 as the first law should apply to each system. Furthermore P(ei) should be a function of P(ei/T) where T is a parameter which behaves as 1/LL because ei is proportional to 1/LL. In this way any function of P(ei) remains unchanged under a quantum adiabatic transformation for which i’s stay constant, but L changes. As a result, the entropy at the quantum level seems to follow from such arguments and not from large N approximations of factorial counting approaches. In large N situations it is argued that P(ei)=C(T)exp(-ei/T) follows from maximizing S with respect to the form of P(ei) subject to constant average energy. For small N situations (e.g. the 2 level example) one may solve Sum over i ei d(a(i)a(i)) = T dS directly without maximization. The maximization approach follows directly from dE=TdS (work=0) because dE= Sum ei dP(ei) in such scenarios. This is of the form of a Lagrange constraint This has been pointed out in Part II. In this note, we argue that the maximization of entropy is not needed at all. For dE=Sum over i ei dP(ei) and the ln form of entropy P(ei) is the inverse function of ln. Thus we argue that linearity and not large N arguments are enough to obtain Shannon’s form of entropy and that this ln form suggests P(ei/T) = C(T)exp(-ei/T) if dE=Sum over ei dP(ei)= TdS (for dW=0). Thus maximization of entropy subject to constant average energy is also not needed (although it does follow mathematically from dE=TdS.)
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.
In (1), we noted that a Gaussian wavefunction may be a solution to the time-independent Schrodinger equation (e.g. the ground state solution of the quantum oscillator) and simultaneously a solution of maximizing spatial Shannon’s entropy - density(x) ln(density(x)) subject to the constraint V(x) xx constant, where density(x) =W*(x)W(x) with W(x) being the wavefunction. In (2), which deals strictly with classical probabilities and the Maxwell-Boltzmann factor n(ei) = exp(-ei/T), we argued that this function is equivalent to: ln(n(ei)) = -ei/T. This in turn is equivalent to the maximization of ln(n(ei)!) with the constraint ei n(ei) which is part of: ln( N!/ Product over i n(ei)!) with the full constraint Sum over i ei n(ei)=Eave. Here n(ei)=N p(ei) where p(ei) is a probability and N a very large number. Thus a function of exp( something) may indicate that “something” * density(x) is a constraint with -density ln(density) being Shannon’s entropy. Quantum mechanics for a bound system yields a real wavefunction W(x) such that density(x)=W(x)W(x)=probability(x). Thus maximization of Shannon’s entropy - density(x) ln (density) subject to a constraint: density(x)B(x) should correspond to a solution for density of exp( B(x)). B(x) as part of the constraint, however, should represent a physical constraint. Thus if there is a solution to the Schrodinger equation of the form exp(.5B(x)) and B(x)density(x) represents the physical constraint in the system, classical maximization of probability solves the quantum problem in such a case. The above ideas may be applied to the quantum oscillator ground state and essentially repeat what is done in (1). We, however, try to generalize them to see if they may apply to a free quantum particle. In this case the probability used is exp(ipx) which is complex. It corresponds to P(x)=exp(ipx)exp(-ipx)=1 which is a uniform distribution which has maximum lack of information.
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.
In Part I we argued that both the first law of thermodynamics dE=TdS-dW and the form S/(-k) = Integral dp dx P(x)P(p) ln{P(x)P(p)} = Integral P(x)ln(P(x)) + Integral P(p) ln(P(p)) may be established at the level of a pure quantum bound state with P(x)=W(x)W(x) and P(p)=a(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx). We further argued that S=Sp+Sx in order that L, the length of a box with infinite potential walls, should disappear (from S) so that a quantum adiabatic transformation where n=energy level remains constant is isentropic. In fact, we argued that the P(x)ln(P(x)) form follows from this rule and an assumption of additive entropies Sp and Sx. As a result, probability at the pure bound state level is not based on the maximization of entropy as P(x)=W(x)W(x) and P(p) =a(p)a(p) and W(x) follows from the solution of the time-independent Schrodinger equation. In this note we examine the case of two level bound state system (analyzed in (1)) to see how the thermodynamic temperature behaves in such a case and examine the maximization of entropy. T is already present in the pure energy state picture (TdS), but for two levels or more, it plays a different role in that P(ei) where ei is the ith energy level becomes a function of T which enters as a Lagrange multiple during the maximization of S with respect to P(ei) with the constraint Sum over i eiP(ei) = Eave. No such maximization of entropy occurs at the pure bound state level. We note, however, that for a quantum adiabatic transformation for a particle in a box with infinite potential walls, P(ei) values must remain constant as L changes. Ei is proportional to 1/LL so a solution is to use ei/T with T is proportional to 1/LL. Thus the parameter T already enters P(ei) in order to ensure that an adiabatic transformation is isentropic. In the pure bound state, one was forced to use P(x)P(p)ln(P(x)P(p)) because P(x) is proportional to 1/L and P(p) to L. In other words in the pure state there are two kinds of probabilities (p and x) and one must ensure that L disappears. For the two level system there is only one type of probability which contains ei(L) so this L must be offset within the probability itself i.e. by using T. Thus thermodynamical and statistical mechanical behaviours which seem to be present at the pure quantum bound state level have different features for multiple level systems even though the first law of thermodynamics remains the same as does the form of entropy. What seems to change is the role of T temperature. We try to examine these features in this note. In particular we also find that maximization of entropy subject to constant energy which is usually linked to high N scenarios (i.e. factorial calculation in the high N limit) follows from the first law of thermodynamics with dW=0. Thus it holds even for one quantum particle with two energy levels (particle in a box with infinite walls). This, however, makes use of the form E=Sum over i ei P(ei). Such a form does not hold for example for a pure bound state.
Extremization of entropy, usually subject to an average energy constraint,(Maxwell-Boltzmann gas) is often associated with maximization of the number of microstates N gas particles may represent. The math procedure of extremization (variational calculus) is then associated with the ln of a factorial expression which reduces to n(ei)ln(n(ei)) using Stirling’s approximation. The central limit theorem, on the other hand, states that the distribution of sample means of a random variable approaches a Gaussian, i.e exp(-pp/(2sig sig)). Here sig sig is the standard deviation. For a random variable, the important or characteristic information is the standard deviation associated with the average of the square pp. We suggest here that an alternative way of looking at maximization of entropy (which is a stability statement we argue) other than ln of permutations is to consider the stability of the variation of the variance of pp subject to a change in the probability, with pp written in terms of P(p). Average kinetic energy in a statistical problem represents such a variance and should be invariant under changes of the probability subject to an average kinetic energy constraint, we suggest. How does this idea apply to quantum mechanics? In quantum mechanics, one has a statistical function exp(ipx) and for bound states creates W(x)=wavefunction = Sum over p a(p)exp(ipx). a(p)a(p) is formally the probability for p, but this does not mean that a(p)a(p) is a random variable because one has the constraint: Sum over p a(p)a(p) pp2/m + Integral dx V(x) W(x)W(x) = En. In the case of a harmonic oscillator, however, one knows a priori that the two summands are each equal to En/2. Furthermore there are a set of En values, starting at a ground state and increasing. All En values, except the ground state, are metastable. Thus the ground state is a special state. Given that Sum over p a(p)a(p)pp2/m = Eo/2, a priori, one may consider that all of the information of a(p)a(p) is associated with Eo. This is exactly the case for a Gaussian from the central limit theorem. Alternatively, from information theory, information = ln(probability) which yields the same result. If this is the case, then pp/2m a(p)a(p) written strictly in terms of P(p)=a(p)a(p) should be stationary upon changing P(p) subject to the E/2 constraint. This special situation, however, only holds for the ground state oscillator. What about average potential energy for the ground state oscillator? If the probability for p (momentum) is a Gaussian due to the Central Limit theorem, this means that the standard deviation is the information (and is linked with randomness). Taking a Fourier transform of a(p) cannot add more information to the picture, we argue. W(x)W(x) should also be a Gaussian, as it is. As a result, xxW(x)W(x), written in terms of P(x)=W(x)W(x) subject to the same Eo/2 constraint should represent a maximization or stationary situation, we argue.