Classical mechanics in the ideal gas scenario and quantum mechanics make use of a set of energy levels which a single particle may occupy. Thus the problem becomes to determine the probability associated with ei i.e. P(ei). In this note we take the approach that relative probabilities depend on the energy gap in question. For example, consider a quantum two level system where ei values are provided by the time-independent Schrodinger equation. From the point of view of statistics, these are simply a set of ei values. Their associated physics is contained in the Schrodinger equation. For the quantum two level system one must find the relative probabilities PR(e1) and PR(e2). Next, imagine a quantum system with many levels and assume for the sake of argument there are levels e1,e2 and ei,e(i+1) such that e2-e1=e(i+1)-ei. If the mechanism is the same for jumping from e1 to e2 and from ei to e(i+1) (i.e. absorbing a photon of the same energy), one might argue that the relative probabilities PR(e2)/PR(e1) = PR(e(i+1))/PR(ei). From this one may obtain the Maxwell-Boltzman result exp(-ei/T)). Note that levels do not need to be adjacent. For example in a large level system one might consider e1 and e3 and any ei and ej such that e3-e1 = ej - ei. In some situations this may need to be modified (e.g. fermions and bosons have enhancements and there may be level related issues), but as a first step we argue that insisting on relative probabilities being dependent on ei-ej has important consequences. Furthermore, in such a case the normalization constant for these relative probabilities i.e. C(T)= Sum over i exp(-ei/T) is important. Taking ln(C(T) and then taking the derivative d/dT yields an expression proportional to the average energy. An expression for dE/dT dT, however, would lead to a second order differential equation in ln(C(T)). If one introduces a new function S such that E-TS = -Tln(C), then one may show that dE/dT = T dS/dT i.e. entropy S emerges.
Classical mechanics in the ideal gas scenario and quantum mechanics make use of a set of energy levels which a single particle may occupy. Thus the problem becomes to determine the probability associated with ei i.e. P(ei). In this note we take the approach that relative probabilities depend on the energy gap in question. For example, consider a quantum two level system where ei values are provided by the time-independent Schrodinger equation. From the point of view of statistics, these are simply a set of ei values. Their associated physics is contained in the Schrodinger equation. For the quantum two level system one must find the relative probabilities PR(e1) and PR(e2). Next, imagine a quantum system with many levels and assume for the sake of argument there are levels e1,e2 and ei,e(i+1) such that e2-e1=e(i+1)-ei. If the mechanism is the same for jumping from e1 to e2 and from ei to e(i+1) (i.e. absorbing a photon of the same energy), one might argue that the relative probabilities PR(e2)/PR(e1) = PR(e(i+1))/PR(ei). From this one may obtain the Maxwell-Boltzman result exp(-ei/T)). Note that levels do not need to be adjacent. For example in a large level system one might consider e1 and e3 and any ei and ej such that e3-e1 = ej - ei. In some situations this may need to be modified (e.g. fermions and bosons have enhancements and there may be level related issues), but as a first step we argue that insisting on relative probabilities being dependent on ei-ej has important consequences. Furthermore, in such a case the normalization constant for these relative probabilities i.e. C(T)= Sum over i exp(-ei/T) is important. Taking ln(C(T) and then taking the derivative d/dT yields an expression proportional to the average energy. An expression for dE/dT dT, however, would lead to a second order differential equation in ln(C(T)). If one introduces a new function S such that E-TS = -Tln(C), then one may show that dE/dT = T dS/dT i.e. entropy S emerges.
Classical statistical equilibrium appears to be based on a no-flow situation with a balance of pressure (including -dV/dx in the case of a potential present). It may be noted, however, that like quantum mechanics, a single particle has different probabilities of being at one x versus another. Furthermore, the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions may be obtained using reaction balance which is linked to conservation of energy i.e. g(f(ei))g(f(ej)) = g(f(el))g(f(ek)) where ei+ej=el+ei=e. Taking ln of the former expression and linking to the latter yields f(ei), the probability distribution. g=1 for the MB case, f(e)/(1-f(ei)) for the FD and f(e)/(1+f(ei)) for the BE. In classical mechanics, one has the conservation equation: p(x)*p(x)/2m + V(x) = E. In quantum bound systems one has: KE(ave)(x) + V(x) = E which appears to be very similar, although it is based on ensemble averages and there is a momentum distribution P(p/x) at each x. A question which arises is: What is the “conservation of energy” equation for a particular p of the P(p/x) distribution? In this note, we argue the “potential” term associated with a component p*p/2m is related to the part of the potential which contributes to the “creation” of p. In particular, we consider V(x)= Sum over k V(k) exp(ikx) where each k term renders a kick of momentum k. One may create a momentum p using k and p-k for each value of k. We further argue that: p*p/2m P(p/x) + contributing portion of V(x) = E P(p/x) ((1)) for all p. Thus, in quantum equilibrium, which involves a flow, there is a “constant E” type of equation i.e. ((1)), different in form but similar in idea, to that of classical statistical mechanical equilibrium. In previous notes, we referred to this as cycling, but in this note we wish to consider it in terms of dynamics. We further note that in both classical statistical mechanics and quantum mechanics, the energy-probability p equation makes use of the overall average energy which is bT for the classical case and E for the quantum. (b is constant)
A Maxwell-Boltzmann gas consists of particles which undergo physical collisions which are associated with the idea of causality, but we argue that reaction balance: e1+e2=e3+e4 and n(e1)n(e2)= n(e3)n(e4) represent steady stream energy-probability conservation and so one need not actually think in terms of direct collisions and causality i.e. one may just as well write: n(e1)n(e2)=n(e1+e2). In the case of quantum mechanics, we argue, as we have done in previous notes, that the free particle form follows from the Lorentz invariant A= -Et+px. In a rest frame, there is energy, but no net momentum. We thus argue for a physical separation of the -Et and px terms. We further suggest that energy is associated with a resonance in time and p momentum with one in space i.e exp(-iEt) and exp(ipx). Given that A=-Et+px is also the relativistic/nonrelativistic classical action, we suggest that these represent probability resonances associated with constant p motion. As a result, given that p is associated with impulse, we suggest that an interaction means forming a new resonance i.e. p1->p2 implies exp(ip1 x)--> exp(ip2 x). As a result, interactions take place in length regions of hbar/p. If this value is very small, as it is in the classical world, then it appears that the resonances occur at points. In the quantum world, it is the opposite. If one wishes to treat quantum mechanics in a statistical manner, one needs to establish an energy-probability balance. Even in classical mechanics, impulses occur that change p, but now one has resonances which occur over a wavelength long piece of space. Furthermore a given p may be anywhere, so one thinks in terms of a(p)exp(ipx) probabilities. We try to show that a probability-energy balance is equivalent to the Schrodinger equation. Thus we argue that quantum mechanics uses a p-probability-energy balance for the time-independent Schrodinger equation and so does not necessarily address causality and physical dynamics.
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 a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by exp(ipx).
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.
In classical statistical mechanics, although one has averages of a system, a single particle within the system has properties which are sharply defined at any x,t, even though these properties change stochastically, i.e. through interactions. A quantum free particle, on the other hand, has information based on the Lorentz invariant -Et+px, multiplied by i to create periodicity which indicates an inherent information, not one based on the outcome of trial runs or interactions. The probability is exp(-iEt+ipx). If E and p are considered as constant, then there are independent probabilistic fluctuations in x and t, which means there may be corresponding probabilistic interactions, both impulse ones (one dimensional reflection/refraction, n-slits) and temporal ones (Rabi frequency). In such a case, one has interference and the form exp(ipx) or exp(-iEt) governing the statistics, not a maximization of entropy linked to interaction/outcome results. Nevertheless, the quantum particle interacts with fields, it seems, which give rise to p or E changes. It seems the key idea is that one has stochastic uncertainty before an interaction occurs and hence interference. The alternative is that such stochastic uncertainty does not exist before a specific interaction, i.e. the inherent probability exp(-iEt) or exp(ipx) has no effect on the interaction. This is not to say that there cannot be probabilities before the interaction. In a Maxwell-Boltzmann gas, there are probabilities P(e1) and P(e2) for particles with e1 and e2, but a specific e1 and e2 have no uncertainty. The uncertainty is in the outcome of the reaction. Another example is a two state electron in an atom which absorbs and emits photons. The rules for time evolution are that the ground state is represented by exp(-iEot), but at some stochastic t value, it changes to exp(-iE1t) and vice versa. The ground state absorbs a photon and the excited emits. The two states are clearly defined by the differing interactions which govern their time dependence which is taken to be simply the changing of Eo to E1 and vice versa. This problem is then based on probabilities based on outcomes of an interaction and maximization of entropy subject to a constraint is employed. There is no initial interference.