In (1) the ideal gas result P V (to the power gamma) = constant ((1)) where gamma=cp/cv=5/3 is applied to a pure quantum bound state for a particle in a box with infinite potential walls and it is seen that ((1)) is a constant just as in the case of an ideal gas. This, however, depends on the result PV=nRT. {An ideal gas, however, follows from a Maxwell-Boltzmann distribution which is linked to the equipartition theorem. The value cp/cv=5/3 reflects this. }Furthermore, T and V are distinct physical variables for an ideal gas. For a particle in a well with infinite potential walls, E=constant/LL. The energy depends only on L whereas in an ideal gas it only depends on T i.e. E=3/2 T. In this note we try to analyze why the ideal gas relationship ((1)) holds for a particle in a box with infinite walls. We also try to link the relationship E proportional to 1/LL to scaling features which ensure that S=Sp(n)+Sx does not depend on L allowing for an adiabatic transformation to keep entropy constant. (This holds for both a particle in a box and a harmonic oscillator.) We use the thermodynamical law dE=TdS-PdV and find that PV=nRT only holds in the one dimensional gas i.e. PL= constant T. In such a case we are able to justify ((1)), but not in the 3-D situation.
In (1) the ideal gas result P V (to the power gamma) = constant ((1)) where gamma=cp/cv=5/3 is applied to a pure quantum bound state for a particle in a box with infinite potential walls and it is seen that ((1)) is a constant just as in the case of an ideal gas. This, however, depends on the result PV=nRT. {An ideal gas, however, follows from a Maxwell-Boltzmann distribution which is linked to the equipartition theorem. The value cp/cv=5/3 reflects this. }Furthermore, T and V are distinct physical variables for an ideal gas. For a particle in a well with infinite potential walls, E=constant/LL. The energy depends only on L whereas in an ideal gas it only depends on T i.e. E=3/2 T. In this note we try to analyze why the ideal gas relationship ((1)) holds for a particle in a box with infinite walls. We also try to link the relationship E proportional to 1/LL to scaling features which ensure that S=Sp(n)+Sx does not depend on L allowing for an adiabatic transformation to keep entropy constant. (This holds for both a particle in a box and a harmonic oscillator.) We use the thermodynamical law dE=TdS-PdV and find that PV=nRT only holds in the one dimensional gas i.e. PL= constant T. In such a case we are able to justify ((1)), but not in the 3-D situation.
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.
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.
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.)
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.)
In a previous note (1) we argue that the entropy for a pure quantum bound state is given by S=Sx+Sp where Sx = -k{ Integral dx Wn(x)Wn(x)ln(WnWn) + Integral dp an(p)an(p) ln(an an) }. It was shown that for any energy level pure bound state for the case of a particle in a box with an infinite potential or for an oscillator S does not depend on the parameters L (length of box) or k (spring constant) or m (mass). Thus a quantum adiabatic change of the parameter leaving the particle in the same quantum state leads to no change in entropy. In (2) a two level quantum Carnot cycle is considered for a particle in a box with an infinite potential. Energy levels n=0 and n=1 are considered.. The wavefunction is a(1)W1(x) + a(2)W2(x). Shannon’s entropy is used namely S=-k Sum over n an(an) ln{|an an|} ((1)) and an expression is found linking an*an to L and T (temperature) for a particle in a box with infinite potential walls. A Carnot cycle with two isotherms and two adiabatic dS=0 parts is considered. No consideration is given to the internal entropy of a bound state eigenstate. In this note we try to show how the overall entropy changes if the internal bound energy eigenfuntion entropy ((1)) is considered and also show how an*an(T,L) changes. We find that a1*a1 changes as do the overall values of Q (heat) and W(work) for the Carnot cycle, but that the efficiency does not.