It is known that a quantum bound state wavefunction may have humps which are linked to the energy state n (e.g. a particle in a box with infinite walls or an oscillator). W(x)W(x) (bound) represents a probability distribution. For a fixed variance, a Gaussian distribution maximizes entropy. This is the wavefunction of the ground state of an oscillator, but adding structured phonons not only increases the energy of the state (i.e. moves it from the ground to an excited level), but gives structure (humps) to the distribution. We argue that these humps represent information about the way in which the state may decay. Thus quantum bound states differ from maximized entropy states subject to an overall constraint linked to say energy etc. There seems to be a correlation between higher energy and spatial information which ultimately follows form the momentum-spatial part of the classical action: px (from -Et+px) being held constant.
It is known that a quantum bound state wavefunction may have humps which are linked to the energy state n (e.g. a particle in a box with infinite walls or an oscillator). W(x)W(x) (bound) represents a probability distribution. For a fixed variance, a Gaussian distribution maximizes entropy. This is the wavefunction of the ground state of an oscillator, but adding structured phonons not only increases the energy of the state (i.e. moves it from the ground to an excited level), but gives structure (humps) to the distribution. We argue that these humps represent information about the way in which the state may decay. Thus quantum bound states differ from maximized entropy states subject to an overall constraint linked to say energy etc. There seems to be a correlation between higher energy and spatial information which ultimately follows form the momentum-spatial part of the classical action: px (from -Et+px) being held constant.
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.
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.
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 (1) the Wigner transform of an oscillator is calculated and the high n energy level limit taken using the WKB approximation yielding a result proportional to delta(p - p(x)) + delta(p+p(x)) where p(x) = sqrt(2m(E-.5kxx)). The Wigner transform involves quantum wavefunctions which for bound systems include +/- p components which interfere i.e. a(p)(exp(ipx) + exp(-ipx) for a(p)=a(-p). Classically, a bound particle moves to the right during one time interval and to the left during the other so there is no interference of +/-p. The quantum wavefunction retains +/-p interference for all energy levels and so does not portray the full physical information of the bound system i.e. leads to the two delta functions. For high n (energy level), crests and troughs are very narrow compared to the length of the system, approximating dx units of length. Thus, uncertainty in time is only within these tiny crests/troughs, otherwise there is a distinct time position relationship as in classical physics, but the wavefunction W(x) does not show it. How then may the wavefunction be interpreted? We argue the time independent wavefunction, say for a bound state, represents an averaged steady stream picture which is often not the physical picture. For example, a bound particle is not a steady stream average nor is a finite number of photons hitting an interface of media with different indices of refraction/reflection. Nevertheless, the steady stream approach with time removed may be used to solve the problem i.e. ultimately find a one dimensional spatial density (which averages out the original two dimensional probability of exp(ipx), energy levels etc. For instance, the energy levels of a bound system do not depend on following the particle in time. We raise this issue because we think it may have implications on the calculation of entropy in bound single particle quantum systems. In general, a spatial Shannon’s entropy is defined as: - W(x)W(x) ln[W(x)W(x)] (one dimensional problem). W(x) (wavefunction) is real because p/-p pieces interfere in a steady stream picture. A single bound particle, however, is not a steady stream picture. exp(ipx) is a dynamic probability equal to cos(px)+i sin(px) to indicate the direction of motion. Simply because exp(ipx)+exp(-px) removes the complex portion (i.e. for a steady state picture with a(p)=a(-p)) does not mean that this represents the full physical picture especially as energy levels become high and crests/troughs become narrower with respect to the overall system length it seems there is no justification for the steady stream interference picture.. In other words, using a real W(x) in a Shannon’s entropy expression treats one dimensional probability W(x) = bound wavefunction, but this is not the full probability in the picture, we argue. Full probability is two dimensional and demonstrates direction for many energy levels (especially high ones). For a free particle exp(ipx), cos(px) and sin(px) are essentially the same distribution. Thus cos(px)cos(px) describes the probability distribution in x. For W1(x)=Sum p>0 a(p)exp(ipx), Real(W1(x)) is not equal to Imaginary(W1(x)) because one does not have a simple shift of the real distribution to obtain the imaginary one. The real distribution is chosen to represent probability because the interference of exp(ipx), exp(-ipx) terms removes the imaginary piece, but it is removed only on average. If physically exp(ipx) and exp(-ipx) should not interfere, then one may need to take some kind of average of Real(W1(x))Real(W1(x)) and Imaginary(W1(x))Imaginary(W1(x)) and use this in Shannon’s entropy.
The spatial and momentum wavefunctions of a ground state quantum oscillator are Gausians in x and p respectively. A Gaussian is also the distribution which maximizes Shannon’s entropy - Sum over i P(i) ln(P(i)) for a fixed variance. Usually one would use average energy as a constraint, but for the nonrelativistic oscillator KE = pp/2m and PE=.5kxx so such a constraint is linked to the variance of x and p. In this note we consider the following questions? Why is the ground state of the quantum oscillator (i.e this form) the only quantum solution which matches a maximum entropy case (for a fixed variance)? Is there a link to a lack of information? Why do higher oscillator states represent non-maximum distributions for fixed variances? Can this also be linked to information? Here we apply the term information to actual information in the system not ln(P(i)).
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.