There is a well-known analogy between statistical and quantum mechanics. In statistical mechanics, Boltzmann realized that the probability for a system in thermal equilibrium to occupy a given state is proportional to \(\exp(-E/kT)\), where \(E\) is the energy of that state. In quantum mechanics, Feynman realized that the amplitude for a system to undergo a given history is proportional to \(\exp(-S/i\hbar)\), where \(S\) is the action of that history. In statistical mechanics, we can recover Boltzmann's formula by maximizing entropy subject to a constraint on the expected energy. This raises the question: what is the quantum mechanical analogue of entropy? We give a formula for this quantity, which we call ``quantropy''. We recover Feynman's formula from assuming that histories have complex amplitudes, that these amplitudes sum to one and that the amplitudes give a stationary point of quantropy subject to a constraint on the expected action. Alternatively, we can assume the amplitudes sum to one and that they give a stationary point of a quantity that we call ``free action'', which is analogous to free energy in statistical mechanics. We compute the quantropy, expected action and free action for a free particle and draw some conclusions from the results.
There is a well-known analogy between statistical and quantum mechanics. In statistical mechanics, Boltzmann realized that the probability for a system in thermal equilibrium to occupy a given state is proportional to (\exp(-E/kT)), where (E) is the energy of that state. In quantum mechanics, Feynman realized that the amplitude for a system to undergo a given history is proportional to (\exp(-S/i\hbar)), where (S) is the action of that history. In statistical mechanics, we can recover Boltzmann's formula by maximizing entropy subject to a constraint on the expected energy. This raises the question: what is the quantum mechanical analogue of entropy? We give a formula for this quantity, which we call quantropy''. We recover Feynman's formula from assuming that histories have complex amplitudes, that these amplitudes sum to one and that the amplitudes give a stationary point of quantropy subject to a constraint on the expected action. Alternatively, we can assume the amplitudes sum to one and that they give a stationary point of a quantity that we call free action'', which is analogous to free energy in statistical mechanics. We compute the quantropy, expected action and free action for a free particle and draw some conclusions from the results.
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 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 (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.
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.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
Classical statistical equilibrium, for a Maxwell-Boltzmann gas for example, is based on localizing the gas to a certain volume and having no average changes in time. In other words, maximization of entropy subject to the average energy constraint sum over i ei P(ei) is linked with these two ideas. Ei (kinetic energy for no V(x)) is a conserved quantity in reactions and overall energy represents a quantity which does not change in time consistent with a center-of-mass which does not change and average values which do not change in time. This approach is associated with a specific kind of statistics, namely that of the constituents of the system. As a result, one obtains the Maxwell-Boltzmann distribution exp(-ei/T) where ei is an extensive property of a particle (or subset of the system) and T an intensive property linked with the whole system. Entire systems, however, may interact. For example, a N particle system with T1 may achieve equilibrium with a N system with T2 yielding a new average (T1+T2)/2. The probabilities exp(-ei/T), however, are for subcomponents of a system. One could argue that momentum is an additive (vector) quantity which is conserved in collisions, but the average momentum of the system is 0 which does not allow one to describe increases in average p and -p which may be described using energy, i.e. through T temperature. This begs the question whether momentum may be associated with maximization of entropy subject to a constraint. In (1), we argued that there were flow probabilities in quantum mechanics linked with exp(-iEt) and exp(ipx), which may be obtained formally through maximization of entropy. Here we argue that while an MB gas describes a localized volume with average values not changing in time, x and t state values change, even if p and pp/2m remain constant for a free particle and it may be possible to have a maximum entropy situation to describe the combination of px and -Et. In other words, we suggest localized versus nonlocalized equilibria. The latter represents a quantum free particle, but one may then try to localize such a particle in a potential. What happens in such a case? One does not switch to a localized maximization of entropy as in the MB gas case, because it is possible to have the potential leave the flow unbroken as argued in (1). In other words, the nonlocalized free particle equilibrium is akin to a particle bouncing back and forth in a box, which in classical mechanics is not considered a statistical process. Here we suggest that the underlying picture is statistical meaning that the classical mechanics approach seems to be an approximation. We consider both a quantum particle trapped in an infinite potential well and the ground state of an oscillator to examine ideas of maximizing entropy beyond exp(ipx). A key common feature of both types of equilibrium is the product rule P(ei)P(ej) = P(ei+ej) or exp(ip1x)exp(ip2x) = exp(i (p1+p2)x). We consider this feature as well.