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.
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.
Classical statistical mechanical density maximizes entropy subject to constraints i.e. one has the Maxwell-Boltzmann (MB) distribution P(x)P(p) proportional to exp(- pp/2mT - V(x)/T). Quantum mechanics is also a statistical theory, but it is not based on the maximization of entropy. Rather it is based on a conditional probability P(p/x) = a(p)exp(ipx)/W(x) where W(x) =wavefunction = Sum over p a(p)exp(ipx) such that: Sum over p pp/2m P(p/x) + V(x) = E (bound states). As a result, it has classical densities P(x)=W(x)W(x) and P(p)=a(p)a(p), but these in general do not maximize entropy and so do not match the MB distribution. There is a special case, however, namely the ground state oscillator for which the quantum mechanical results P(x) and P(p) are both Gaussians. It is shown in (1) that for a fixed variance, a Gaussian maximizes entropy. Thus, the MB scenario must be a Gaussian for an oscillator as it is, but the quantum result is also a Gaussian and so matches the classical statistical mechanical result. In this note, we show that the time-independent Schrodinger equation collapses to the equality situation in Heisenberg’s uncertainty principle, suggesting that the solution also represents maximum entropy.
For a classical ideal gas or single particle quantum system with energy levels {ei}, both average energy E=Sum over i ei P(ei/T) and Shannon’s entropy S= - Sum over i P(ei/T) ln(P(ei/T) change (for mechanical parameters kept fixed) in terms of d(P(ei/T)). In fact, in this case ln(P(ei/T)) appears as a “constant vector” in dS i.e. dS= - Sum over i ln(P(ei/T)) dP(ei/T). Both dE (mechanical parameter fixed) and dS change as dP(ei/T), thus there are two vectors {ei/T} and {ln(P(ei/T))} which represent constants over ei’s for a small range of dT where T is temperature. In this case, however, the two vectors are strongly linked as P(ei/T)=exp(-ei/T)/C(T) so ln(P) = -ei/T - ln(C). This is why the relationship E-TS = -T ln(C(T)) exists. For a quantum bound state one may write: e = Integral dp a(p)a(p) pp/2m + Integral dx V(x) W(x)W(x) where W(x)=Sum over p a(p)exp(ipx) is the wavefunction. Instead of having vectors of ei’s, one has two vectors, one over x and the other p (momenta) where a(p)a(p) = P(p) and P(x)=W(x)W(x) (bound states). Thus d e thermal = Integral dp pp/2m dPe(p) + Integral dx V(x) dPe(x) ((1)). In other words de thermal changes linearly in terms of dPe(p) and dPe(x) whereas for a classical ideal gas or many ei system it changes linearly in terms of dP(ei/T). Ei is the variable instead of x and p in that case. In other words {pp/2m} is a constant vector in p space and {V(x)} in x space. Mathematically Shannon’s entropy Sp = - Sum over p Pe(p) ln(Pe(p)) and Sx =- Sum over x Pe(x) ln(Pe(x)) also yield constant vectors in their respective spaces namely ln(Pe(p)) and ln(Pe(x)) which are called information. In particular dSp = - Sum over p ln(Pe(p)) dPe(p) then ln(Pe(p)) behaves like a constant vector within a region of small dPe(p) changes. Note: Sum over p dPe(p)=0. One may consider a set of single valued energy levels ei which are very close (imagine a particle in a box with infinite walls) as a region in which dPe(p) changes (or dPe(x) ones) are very small. In this region, ln(Pe(p)) and ln(Pe(x)), information, represents a constant vector statistically defining the small region of ei states. This we argue is the statistical significance of information ln(P). One may note that both de ((1)) adn dSp and dSx change linearly with dPe(p) and dPe(x), but the vectors {pp/2m} and {V(x)} over p and x space and ln(Pe(p)) and ln(Pe(x)) over the same space are constant vectors, but not equivalent except for the ground state of the quantum oscillator for which they become very similar because Pe(p)=exp(-app) and Pe(x)=exp(-xx/a) with a being proportional to 1/ e which plays the same role as temperature in the classical ideal gas case. For other quantum bound systems, the two sets of vectors are different, but the information vectors seem to give a kind of statistical identity to small regions containing very close ei levels for which dPe(p) and dPe(x) are small. Thus, though one may argue for the presence of ln(P) in Shannon’s entropy S=-Sum over i P(i)ln(P(i)) in many ways, it seems that ln(P(i)) is a constant vector in a certain space i for small dP(i) and gives a statistical identity to that small region.
The notion of entropy originates historically in classical physics and is intertwined with thermodynamics. Shannon’s entropy: - Sum over i P(i) ln(P(i)), where P(i) is the probability for i, however, seems to apply to any probability, but in quantum mechanics appears to be particularly linked to spatial and momentum entropy (1) i.e. - Integral dx W*(x)W(x) ln[ W*(x)W(x)] and - Sum over p a*(p)a(p) ln[ a*(p)a(p) ] where W(x)= wavefunction = Sum over p a(p) exp(ipx). In this note we speculate that one may create an entropy from the real part of exp(ipx) i.e. cos(px) as this term is associated with probability within a wavelength even though the “center of mass” of the particle may be moving uniformly at v=x/t=constant and W*(x)W(x)=1 or a constant. The latter implies an entropy independent of p but there is uncertainty in a free particle due to its wavelength which is experimentally manifested. We argue there is no reason to limit entropy to x or p considerations. For a bound state, a pattern of crests and troughs of different lengths and heights may be produced and so again there is a pattern of unnormalized probability distributions with different weights. The absolute value may be taken so negative wavefunction values are not an issue. For both the free particle and the bound state two features emerge. First the shape of the distributions is linked to the curvature which in turn is associated with the kinetic energy. Thus the probability distributions which describe the quantum system are governed by kinetic energy. Secondly, a pattern of distributions, both negative and positive, ensures that exp(ipx) is perpendicular to other exp(ip1 x) functions and Wn(x) is perpendicular to other Wn1(x)’s. In other words energy identifies the statistical state which exists throughout space and one cannot confuse the identity of one state with another. Thus they are orthogonal throughout space and probability distributions (linked to entropy) seem to enforce this. The inner product of W*(x) and W(x) is like an inner product of a product probability of a state and its time reversed state. We suggest one should be able to define a Shannon’s entropy expression for both |cos(px)|/C1 from exp(ipx) over a half wavelength where C1 is a normalization factor and |W(x)|/C2 roughly over the length over which it has humps i.e. roughly the classical turning points for an oscillator for example. This entropy is not additive and does not map to classical entropy in the large n (energy level) limit. Furthermore it may not be linked to thermodynamics because it is not linked to x or p. Nevertheless, it might be used to describe uncertainty within the quantity system and equilibrium/stable scenarios which are observed in nature. For example, different Wn(x) states are statistically stable states, but one may knock out the particle and it will be in an exp(ipx) state which is also stable. Furthermore, a photon may excite Wn1(x) to Wn2(x), thus this new entropy seems to be linked to stable states allowed in nature. The “price to pay”, however, seems to be uncertainty in wavelength regions for a free particle and the crest/trough regions for a bound state. We also consider some properties of a similar entropy for bound states.
In previous notes, we suggested that quantum bound states may be analysed in terms of the conditional probability P(p/x)= a(p)exp(ipx)/W(x) (where W(x) is the wavefunction) which allows for the calculation of average kinetic energy at each point x i.e. [Sum over p pp/2m P(p/x)]/ W(x). This added to V(x) yields E (average energy) at each x and yields the time-independent Schrodinger equation). We argued that probability is handled at the level of W(x)=Sum over p a(p) exp(ipx). Measurements (even internal) may knock a particle from one p1 state to another so another set of probabilities (P(x) and P(p)) arise, P(x) being based on: a(p1)exp(i p1x) exp(i p2x). Averages of f(p,x) follow from P(x) f(p,x) P(p/x). Thus, there are two sets of probabilities in quantum mechanics. Entropic dynamics (1) appears to begin with the probability P(x) and considers a transition probability P(x1 |x) with x1-x= an infinitesimal quantity. A product of transition probabilities is then taken to allow for finite x1-x differences. It is stated in (1) that there exists information about x but not p. The maximization of entropy density may be used to determine the form of P(x1 | x). A “drift velocity” is also included (which is later shown to be related to the phase of the wavefunction). Ultimately, the Schrodinger equation is also obtained. We argue one may use P(p/x) to also calculate P(x1 | x) and compare this result with that obtained through the maximization of entropy density. A type of drift velocity appears naturally in this approach. We argue that the form a(p)exp(ipx) already incorporates statistical ideas (i.e. maximum entropy) in that the modulus of exp(ipx) is identical for all x, endowing all x points with a “kind” of equivalent status. exp(ipx) is a function of p and x because one must calculate average kinetic energy at each x and this must change with x i.e. KEave(x) + V(x) = E. Thus, we think there are strong similarities between entropic dynamics and the approach of using P(p/x). There are also some differences in that the coefficient A of delta(x)delta(x) in P(x1 | x) = 1/Z exp{ A delta(x)delta(x) + B d Phi/dx delta(x)} is treated as independent of x in entropic dynamics, but is a specific function of x (related to d ln(W)/dx d ln(W)/dx + 1/W d/dx d/dx W) in the P(p/x) approach.
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.
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.)