In a previous note (1) we suggested the existence of a quantum entropy associated with a free quantum particle with momentum p and wavefunction exp(ipx) due to the presence of a wavelength. The usual approach of using P(x)=W*(x)W(x) in Shannon’s entropy does not capture this entropy as it yields a constant for all p values. Thus we suggested using |cos(px)| C where C is a normalization constant over half a wavelength. This value may be used in Shannon’s entropy formula integrated over a half wavelength to give an entropy per crest. We then extended the idea to a quantum bound state. In this note we wish to examine the relationship of this entropy to momentum i.e. energy flow. In (1) we noted that this entropy does not satisfy the first law of thermodynamics. Here we argue that it also does not satisfy the second law using ideas from refraction. Finally we note that quantum thermodynamical calculations using temperature seem to ignore this entropy. In fact, these calculations ignore the usual P(x)=W*W or P(p) = a*(p)a(p) Shannon’s entropies for a pure state suggesting that only some “uncertainty” is included in typical entropy calculations while other uncertainty may apparently be ignored.
In a previous note (1) we suggested the existence of a quantum entropy associated with a free quantum particle with momentum p and wavefunction exp(ipx) due to the presence of a wavelength. The usual approach of using P(x)=W*(x)W(x) in Shannon’s entropy does not capture this entropy as it yields a constant for all p values. Thus we suggested using |cos(px)| C where C is a normalization constant over half a wavelength. This value may be used in Shannon’s entropy formula integrated over a half wavelength to give an entropy per crest. We then extended the idea to a quantum bound state. In this note we wish to examine the relationship of this entropy to momentum i.e. energy flow. In (1) we noted that this entropy does not satisfy the first law of thermodynamics. Here we argue that it also does not satisfy the second law using ideas from refraction. Finally we note that quantum thermodynamical calculations using temperature seem to ignore this entropy. In fact, these calculations ignore the usual P(x)=WW or P(p) = a(p)a(p) Shannon’s entropies for a pure state suggesting that only some “uncertainty” is included in typical entropy calculations while other uncertainty may apparently be ignored.
In Part I we suggested that a free quantum particle with wavefunction exp(ipx) should have a spatial Shannon’s entropy associated with the probability cos(px)/C1 over half a wavelength, where C1 is a normalization constant. This new entropy captures the idea of uncertainty due to the wavelength even though the constant velocity suggests uniform motion i.e. W*(x)W(x) = constant. This idea was extended to a bound state and yields different results than the usual P(x)=W*(x)W(x) and P(p) =a*(p)a(p) based Shannon’s entropies where W(x)=wavefunction = Sum over p a(p)exp(ipx). In this note we try to determine some properties of this new entropy. First we consider the high n limit and use a classical approach introduced in (2). Next we expand the bound state entropy to include both spatial and momentum entropies and show that the sum follows from the product probability |W(x)|/C1 |a(p)|/C2 where 1/C1 and 1/C2 are normalization constants. We also examine quantum adiabatic processes. For P(x) = W*(x)W(x) and P(p)=a*(p)a(p), one finds that such a process leads to Sp+Sx remaining unchanged for L changing adiabatically for a box with infinite potential walls or for k the spring constant changing adiabatically in an oscillator. This follows from the fact that W(x/L) or W(x .5sqrt(km) ) (oscillator), from the normalizations Integral P(x)dx= 1 and Integral P(p)dp =1 and from exp(ipx) which shows that p scales in reciprocal manner of x. Thus we argue that these properties still hold for the entropies defined using |W(x)| and |a(p)| as probabilities. In other words Sp+Sx for the new quantum entropy does not change under a quantum adiabatic transformation. Finally we note that the new entropy is not compatible with the first law of thermodynamics.
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.
The von Neumann entropy definition is -Tr ( d ln(d)) and is linked to: = Tr (d A) ((1)). Here d is a matrix as is A i.e Aij = . As a result d is often not diagonal. It seems further that the idea of entropy is linked to an operator (2). In other words, one might have certain probabilities linked to the operator P momentum and a different set to X. For a diagonal d, von Neumann’s form is the same as Shannon’s entropy, but uses the eigenvalues of d. For a pure quantum bound state Wn(x), it seems that one considers the two operators X and P and writes two entropies using Shannon’s entropy form: - Integral dx W(x)W(x) ln(WW) and -Integral dp a(p)a(p) ln(a(p)a(p)) where W(x)=wavefunction= Sum over p a(p)exp(ipx). Here P(x)=W(x)W(x) and P(p)=a(p)a(p). For example in (3), one may see calculations of these “distinct” entropies for a particle in a box with infinite potential walls. X and P are linked to measurables and for a quantum bound system one cannot know exactly x and p at the same time. One may write < KE > for a bound state in two ways: Integral dp a(p)a(p) pp/2m or Integral dx W(x) (-1/2m d/dx dW/dx). Using Shannon’s entropy approach one obtains two different entropy values Sx and Sp. With respect to von Neumann’s entropy, it is zero for a pure state. Although one cannot know x and p at the same time (in terms of a measurement) that does not prevent one from writing x and p together (i.e. both known) in exp(ipx) for the wavefunction of a free particle. In this note we suggest that one might consider product probabilities linked to more than one operator when considering entropy. For example, for <KE> one may write Integral dx P(x) Integral dp pp/2m P(p/x) thus acknowledging that both x and p play a role. In this case, P(x)P(p/x) is the probability to use in Shannon’s entropy as we have shown in (4). In (3) it is suggested that P(x)P(p/x) be used in order to incorporate both variables x and p. Using P(x)P(p) in Shannon’s entropy yields the same overall entropy when integrated over x and p i.e. S=Sx+Sp. One reason that the sum is useful is that it eliminates parameters such as L (box length) for a particle in a box with infinite walls (or k the spring constant for an oscillator. This allows for a quantum adiabatic transformation of a pure state to be isentropic. This follows from a Fourier transformation i.e. W(x)=Sum over p a(p)exp(ipx). The two variables p and x scale with L in the opposite manner i.e. x/L and pL (5). Thus we argue that entropy is not necessarily linked to only one operator, but rather for a pure state is linked to variables which appear in a Fourier transform. Usually a Fourier transform is simply a mathematical procedure, but not in the case of the quantum pure state wavefunction where the variable p is actually momentum. In this note we try to investigate how the quantum relationship between p and x emerges. We argue that it exists at the level of the classical action for a free particle (relativistic or nonrelativistic) A=-Et+px where dA/dx partial =p and dA/dt partial = -E. Thus p and x are variable pairs in classical dynamics and this carries over to classical statistical phase space linked to p and x. We argue that a quantum dynamical link between p and x is also present in A i.e. -id/dx exp(iA) = p exp(iA). In other words p and x are linked dynamically in quantum mechanics as well (for a free particle) with exp(-iEt+ipx) being part of this link. The Fourier series form is already present in this solution as one may create a momentum distribution i.e. Sum over p a(p)exp(ipx). This depends on p and x and both affect any operator based on p and x (or one of these). Thus we expect that probabilities linked to both p and x should appear in an expression of entropy for a quantum bound state.
In a previous note (1) we considered a free particle entropy based on the real part of the wavefunction exp(ipx) i.e. |cos(px)|. |cos(px|) may be normalized over a certain length, just as exp(ipx) is. The idea is that there exists an uncertainty in space within a wavelength which should be reflected in an entropy expression. As p becomes very large, cos(px) becomes very narrow and has many peaks in L. Thus probability →0 so p ln(p)--> 0 using l’Hopital’s rule. This coincides with a free classical particle having zero entropy. In this note, we revisit arguments presented previously, suggesting that A= -Et+px, a Lorentz invariant and also the free particle classical relativistic and nonrelativistic action for v=x/t We suggested in the past that A is responsible for both the wavelength hbar/p and the time period hbar/E. Here we suggest uncertainty within a time period should be considered together with uncertainty within a wavelength. For instance, recent experiments (2) have shown that one may have quantum interference in time as well as space. Thus there is uncertainty for a free particle in both space and time. We add the entropies for these two. Next, we consider the case of a bound system. Such a system creates a crest-trough pattern (with different heights and widths) for its spatial wavefunction, but E, average energy is constant and there is the free particle type exp(-iEnt). We suggest that the system is characterized by uncertainty in space and time with En being a known constant at each x. This suggests that various free particle p (momentum) values combine to create the wavefunction such that -1/2m d/dx d/dx W/W is average kinetic energy which in turn adds to V(x) to create a constant En. We are thus concerned with uncertainty in time and space and not in momentum. In the limit of high energy values, a crest fits within a tiny dx and prms is essentially the classical p value which is considered certain, even though in quantum mechanics the range of a(p) weights hold for all n (W(x)=Sum over p a(p)exp(ipx)). Thus we treat a high n limit as being completely certain and not having entropy. This differs from the approach of using C/v(x), where v(x) is velocity given by .5mv(x)v(x)+V(x)=En, as a probability and calculating Shannon’s entropy using this.
The Schrodinger equation, a partial differential equation in W(x,t) the wavefunction, was developed by Schrodinger from ideas of L. de Broglie and then given the interpretation of probability Probability(x,t)= W*(x,t)W(x,t) by M. Born. In this note we ask whether the form P(x)=W*(x)W(x) together with P(p)=a*(p)a(p) such that W(x)=Sum over p a(p)exp(ipx) follows from quantum thermodynamics. In particular we concentrate on a quantum adiabatic transformation and insist it be one of no entropy change (as in (1)). The quantum spatial density of a particle in a box with infinite potential depends on L as P(x)= 1/L density(x/L) (bound state). Thus Shannon’s entropy simply using P(x) is L dependent. One needs P(x)P(p) in addition to the first term changing by ln(1/L) and the second by ln(L) in order to remove L dependence during an adiabatic expansion. Similar considerations hold for the quantum simple harmonic oscillator (2). Thus the idea of a P(x) and P(p) which are linked (not independent) immediately follows form quantum thermodynamics as does the form of entropy S=Sx+Sp. As a result it seems that if one deals with probability, one must not only consider conditional probability, but also a transformation function f(px) which allows for inverse dependencies on p and x on a parameter. These ideas together with P(p/x)=P(x and p) / P(x) ((1a)) and P*(x/p) = P(x and p) / P(p) lead, we argue, to the form P(x)=W*(x)W(x) and P(p)=a*(p)a(p) where W(x)=Sum over p a(p) exp(ipx). We considered the approach of conditional probabilities in an earlier note (3), but did not give motivation for using them or the form exp(ipx). Rather we brought them in as assumptions, assuming that there should be both a P(p) and P(x). In this note, we argue that they seem to follow from the idea of quantum thermodynamics and a zero entropy change along an adiabatic path i.e. one which does not change the quantum bound state n.
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.