In statistical mechanics, involving quantum objects, one sometimes uses sums if a simple series sum rule exists. For example, in obtaining the Planck distribution 1/ (1-exp(hf/T)) one uses the sum of a geometric series. Otherwise, it seems one uses integrals, arguing that energy levels are very close together. For the Planck distribution with 1/ (1-exp(fon/T)) where n is 1,2,3…, one usually uses an integral, when a sum really applies. In (1), it is argued that such an integral does not yield the same result as the sum for this Planck distribution example. In particular, (1) considers fo as a very small number and shows explicitly that the summation depends on fo, while the integral does not. Given that this applies to the partition function, it carries over to all statistical quantities calculated from the partition function, as noted in (1). Here, we consider other examples. For example, if one tries to replace the geometric series sum used to obtain the Planck distribution, in the first place, with an integral, one immediately has a degeneracy issue. A one dimensional integral over dp (p=magnitude of momentum) yields one answer, while a three-dimensional result Integral 4*3.14 pp dp yields another. The geometric sum, however, does not include the degeneracies of p so it is not clear why one would use a 3-dimensional integral. Even if one uses a one dimensional integral, it does not yield the same result as the geometric sum. In fact, the geometric sum depends on the value of f/T, while the integral does not - it depends on T. Another example is the partition function for a single quantum particle in a box with infinite potential walls (e.g. a Fermi gas). Again an integral is used, but it seems that using p=po a where a is a small number proportional to 1/L, the length of the box and summing should yield an “a” dependent result, which the integral does not give.
In statistical mechanics, involving quantum objects, one sometimes uses sums if a simple series sum rule exists. For example, in obtaining the Planck distribution 1/ (1-exp(hf/T)) one uses the sum of a geometric series. Otherwise, it seems one uses integrals, arguing that energy levels are very close together. For the Planck distribution with 1/ (1-exp(fon/T)) where n is 1,2,3…, one usually uses an integral, when a sum really applies. In (1), it is argued that such an integral does not yield the same result as the sum for this Planck distribution example. In particular, (1) considers fo as a very small number and shows explicitly that the summation depends on fo, while the integral does not. Given that this applies to the partition function, it carries over to all statistical quantities calculated from the partition function, as noted in (1). Here, we consider other examples. For example, if one tries to replace the geometric series sum used to obtain the Planck distribution, in the first place, with an integral, one immediately has a degeneracy issue. A one dimensional integral over dp (p=magnitude of momentum) yields one answer, while a three-dimensional result Integral 4*3.14 pp dp yields another. The geometric sum, however, does not include the degeneracies of p so it is not clear why one would use a 3-dimensional integral. Even if one uses a one dimensional integral, it does not yield the same result as the geometric sum. In fact, the geometric sum depends on the value of f/T, while the integral does not - it depends on T. Another example is the partition function for a single quantum particle in a box with infinite potential walls (e.g. a Fermi gas). Again an integral is used, but it seems that using p=po a where a is a small number proportional to 1/L, the length of the box and summing should yield an “a” dependent result, which the integral does not give.
Shannon’s entropy: -k Sum over i P(i) ln(P(i)) is expressed in terms of P(i) the probability for state i. If entropy exists as a concept separated from the first law of thermodynamics with dWork=0, then it is not clear what set of states i should be taken in quantum mechanics. In classical statistical mechanics, each particle is automatically described by its position and momentum vectors for no internal motion. For example a pure energy eigenstate wavefunction Wn(x) may be written in terms of many different orthonormal mathematical basis sets e.g. Wn(x)= Sum over i bi phi(i)(x) with probabilities bi*bi. Thus one may create many types of entropy expressions using Shannon’s form without any other physical considerations. For example, W(x)=Sum over p a(p)exp(ipx) and one may create Sp which is usually done in the literature. In previous notes we have argued that Sn = Sp(n)+Sx(n). For a particle in a box with infinite potential walls, Sx(n)=Sx. The sum of the two entropies ensures that the parameter L (box size) vanishes so that a quantum adiabatic transformation is isentropic. Thus a consideration other than Shannon’s form has been used to obtain Sn=Sp(n)+Sx for a particle in a box. Furthermore one may argue Shannon’s form even for a pure state based on S=S1+S2 where 1 and 2 represent two systems. A question still remains: Why are Sx and Sp entropies used and not some other set? Furthermore for a system with several energy levels ei one might ask why -Sum over i P(ei)ln(P(ei)) is used as in (1)? In a previous note (2) we considered that Shannon’s form of entropy follows from dE= TdS (dWork=0) with dE=Sum over i ei dP(ei). Furthermore we argued that P(ei) = C(T)exp(-ei/T) is a solution, but so is C(T) exp(-ei/T) exp(Si) where Si=Sx,i + Sp,i for the pure bound state. In other words, specific states related to the physical situation are chosen and entropy constructed from dE=TdS to reflect this. In this note we argue that entropy may be less related to counting arrangements than to considering the issues involved in energy transfers within a system.
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.)
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator exp(+sH), where H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary c-number integral of a c-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian, Heff, which is a function of the parameter s and the c-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function ZN in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for Heff depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying exp(sHeff). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for Heff. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
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 classical statistical mechanics, C exp[ -( mv*v/2 + V(x))/T) represents density i.e. the probability for a particle to have a velocity v and be at x. If applies this probability to Shannon’s entropy formula, one obtains C/T ( mv*v/2 + V(x)) exp[ -( mv*v/2 + V(x))/T) + exp[ -( mv*v/2 + V(x))/T) ln(C). Thus, it seems energy times density + density times ln(C) is the entropy density, a well known result in classical statistical mechanics, although sometimes not explicitly stated. The first term integrated over x is essentially the kinetic energy, while the second integrated over velocity yields the potential energy term. Thus, the first term partial density seems to be a purely momentum dependent kinetic energy piece times probability and the second a purely x dependent potential energy piece. (There is still an integral over velocity for the first term and one over x for the second to obtain the full entropy.) In quantum mechanics, we argue that a(p1)exp(ip1 x) a(p2) exp(ip2 x) serves as the probability where W(x)=Sum over p a(p) exp(ipx) = wavefunction. This leads to an expression for entropy density with two terms, one of which may be immediately integrated over momentum and the other over space. If one wishes to make a link to the classical statistical mechanical case, one may ask what requirement is necessary for the momentum related piece to represent average kinetic energy and the spatial part, average potential energy. The requirement, we find, yields the wavefunction of the ground state harmonic oscillator. For other cases, the partial density is not equivalent to average energy. Thus, we try to establish a link between entropy density and energy density, but require the kinetic energy related term to be strictly a function of momentum and the potential related term strictly one of x. In quantum mechanics, one may form averages at a point x. For example, KEave(x)= [Sum over p p*p/2m a(p) exp(ipx)]/W(x). If one examines a p/-p pair with a(p)=a(-p) (for example), cos(px) terms arise which are positive at some x and negative at others. We argued in previous notes that this represents stochastic flow in the opposite direction from that expected by -dV/dx. In other words, -dV/dx is an average which consists of stochastic force addends which may give momentum hits to the right or left, but which add to yield -dV/dx. As a result, one may have negative contributions to the kinetic energy (and density as well). In this note, we argue one may also have negative contributions to the entropy density due to this phenomenon.
In this note we wish to examine three ideas. First we consider the case of W(x,t)=Sum over n an exp(-iEn t) Wn(x) ((1)) (for n=0,1 as an example) and note that the von Neuman entropy is 0 for such a state. This is pointed out in (1) together with the fact that there exists a von Neumann density operator exp(-H/T) (H=Hamiltonian operator) which yields the same result as Shannon’s entropy S using P(ei) if one uses dQ/T=Sum over n en dP(en)/T = dS. It seems then that there is a contradiction if the usual von Neumann entropy is 0 yet using a density operator = exp(-H/T) yields nonzero results. A resolution is suggested in (2) which deals with information theory and we see if it applies to ((1)). Secondly, we consider the Shannon’s form P(e)ln(P(e)) in more detail. In Part I and in previous notes we argued that ln ensures that S=S1+S2 if one has two systems. We also argued that dE= Sum over n en dP(en) = TdS if dWork =0. In this note, we argue that a stronger idea may be at work, namely the idea that Eave= Sum over n en P(en/T) and TS are essentially the same function except for a function of T. This approach may be used to show that ln is the function associated with entropy. Finally, we note as in Part I that when using either the von Neumann approach or Shannon’s entropy one must know which probabilities to use. For a system with energy levels en, P(n)=P(en/T) is an immediate candidate, but there may be more probabilities in the system related to a particle jumping from one ei to another. In such a case, if one uses these probabilities and then sums over some of them leaving only P(en/T), the form of entropy changes from that of Shannon’s form. (Originally, however, Shannon’s form is used before summing.)