In (1), the spatial quantum Shannon’s entropy: - Integral dx Wn(x)Wn(x) ln[ Wn(x)Wn(x)] ((A)) for a bound state is considered, where Wn(x) is the nth energy eigenstate. In the limit of n very high, quantum entropy is approximated by: - Integral C/v(x) ln{ C/v(x)} dx ((B)) where C/v(x) is the normalized classical entropy with .5mv(x)v(x) + V(x) = En. In such a case, W(x)W(x) still has peaks and zero points, but if resolution in the classical world is low, the integral is approximated by the classical C/v(x) which is said to pass through the peaks of W(x)W(x). In this note, we consider a bound classical particle moving first to the right and then to the left. In such a case, it seems entropy is zero. The argument in (1) is that the time a particle spends in a fixed dx is dx/v(x) and that this should represent an unnormalized probability. We argue this may represent an unnormalized probability for an ensemble of classical systems with the same energy, but different starting points x,t. If one wishes to know the entropy of a single system, we argue it is 0 and not ((B)). The goal of this note is to find a quantum expression for spatial entropy which tends to 0 in the high n limit. Our approach is to introduce a cut off-length dx which is very small with respect to the system length and is also small in the classical world. Next, W(x)W(x) which has humps and zero values is examined. If the base of any hump is within dx, entropy density is set to 0 within this dx, otherwise it is given by ((A)). As humps become narrower, the entropy becomes smaller as more and more dx units are replaced with 0. In this way, we try to study the change in quantum bound entropy as the system becomes more and more classical with higher En values. Thus one has S(dx, n) which may be studied for various dx and n values.
In (1), the spatial quantum Shannon’s entropy: - Integral dx Wn(x)Wn(x) ln[ Wn(x)Wn(x)] ((A)) for a bound state is considered, where Wn(x) is the nth energy eigenstate. In the limit of n very high, quantum entropy is approximated by: - Integral C/v(x) ln{ C/v(x)} dx ((B)) where C/v(x) is the normalized classical entropy with .5mv(x)v(x) + V(x) = En. In such a case, W(x)W(x) still has peaks and zero points, but if resolution in the classical world is low, the integral is approximated by the classical C/v(x) which is said to pass through the peaks of W(x)W(x). In this note, we consider a bound classical particle moving first to the right and then to the left. In such a case, it seems entropy is zero. The argument in (1) is that the time a particle spends in a fixed dx is dx/v(x) and that this should represent an unnormalized probability. We argue this may represent an unnormalized probability for an ensemble of classical systems with the same energy, but different starting points x,t. If one wishes to know the entropy of a single system, we argue it is 0 and not ((B)). The goal of this note is to find a quantum expression for spatial entropy which tends to 0 in the high n limit. Our approach is to introduce a cut off-length dx which is very small with respect to the system length and is also small in the classical world. Next, W(x)W(x) which has humps and zero values is examined. If the base of any hump is within dx, entropy density is set to 0 within this dx, otherwise it is given by ((A)). As humps become narrower, the entropy becomes smaller as more and more dx units are replaced with 0. In this way, we try to study the change in quantum bound entropy as the system becomes more and more classical with higher En values. Thus one has S(dx, n) which may be studied for various dx and n values.
In the literature (e.g. (1)), the expression - density(x) ln(density(x)) is used as Shannon’s spatial entropy for a quantum system (bound or not). For the bound case, the correspondence principle is said to hold for high energy levels (2). In that case, quantum density = W(x)W(x) (W(x)=real wavefunction) is said to be equivalent to the classical density C/v(x) where v(x) the velocity is given by .5mv(x)v(x) + V(x) = E. Classical density yields a curve which passes through peak regions of quantum density humps. In the classical case, particle motion is completely determinate. C/v(x) as a density (probability) is proportional to dt, the amount of time a particle spends passing through a fixed length dx. If the particle moves quickly, it spends little time, and hence there is less time for a measurement (i.e one may miss the particle due to its speed). We argue in this note that such a scenario does not seem to be in keeping with the ideas of Shannon’s entropy where one has probabilities for certain mutually exclusive events at each ti. For example at each ti, one may toss a coin. If it is not “heads”, this means it is tails. For the classical particle in the correspondence principle, each dx region has its own probability proportional to dt = dx/v(x). Thus, each dx is a region of its own in terms of probability. Thus, if Ci measurements of the presence of the particle are made within dx of xi out of N trials, the probability that a measurement is made in dx at xi is Ci/N and the probability that it is not is 1-Ci/N. Thus, unlike Shannon’s picture, various arrangements do not have the same weight or probability. We investigate this scenario in more detail and argue that it does not follow the approach of Shannon’s entropy. We also try to connect with the quantum situation.
Linking quantum mechanics to classical mechanics seems to have been an early goal in quantum theory as represented by the correspondence principle. More recently there have been attempts to link the entropy of both a single bound quantum state and a thermal statistical system of quantum states to classical entropies (1),(2). For the former, the entropy is proportional to Integral dx density(x) ln[density(x)] where density(x)= delta t = 1 / v(x) with v(x) being velocity. Thus, density, which is a probability in Shannon’s entropy equation, is a measure of the change in the value of delta t(x) in this classical entropy. Bound state quantum mechanics is said to be time-independent, but by the correspondence principle (1) the probability of the envelope of the wavefunction i.e. Envelope(W(x))* Envelope(W(x)) = delta t(x) so quantum probability (albeit linked to the envelope) is related to classical time intervals which adjust with x to yield the proper ration v(x)=delta x / delta t(x) with delta x being constant. In this note we argue that the expectation value: Integral W(x) d/dx dW/dx = Integral WW KEclassical(x) for a bound state is, in the high energy limit Integral dt KEclassical(x) i.e. a portion of the action, just as in the classical case (i.e. the correspondence principle). This leads to the association of the higher energy limit of the envelope of W times itself with delta t(x).
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.
For a classical ideal gas or single particle quantum system with energy levels {ei}, both average energy E=Sum over i ei P(ei/T) and Shannon’s entropy S= - Sum over i P(ei/T) ln(P(ei/T) change (for mechanical parameters kept fixed) in terms of d(P(ei/T)). In fact, in this case ln(P(ei/T)) appears as a “constant vector” in dS i.e. dS= - Sum over i ln(P(ei/T)) dP(ei/T). Both dE (mechanical parameter fixed) and dS change as dP(ei/T), thus there are two vectors {ei/T} and {ln(P(ei/T))} which represent constants over ei’s for a small range of dT where T is temperature. In this case, however, the two vectors are strongly linked as P(ei/T)=exp(-ei/T)/C(T) so ln(P) = -ei/T - ln(C). This is why the relationship E-TS = -T ln(C(T)) exists. For a quantum bound state one may write: e = Integral dp a(p)a(p) pp/2m + Integral dx V(x) W(x)W(x) where W(x)=Sum over p a(p)exp(ipx) is the wavefunction. Instead of having vectors of ei’s, one has two vectors, one over x and the other p (momenta) where a(p)a(p) = P(p) and P(x)=W(x)W(x) (bound states). Thus d e thermal = Integral dp pp/2m dPe(p) + Integral dx V(x) dPe(x) ((1)). In other words de thermal changes linearly in terms of dPe(p) and dPe(x) whereas for a classical ideal gas or many ei system it changes linearly in terms of dP(ei/T). Ei is the variable instead of x and p in that case. In other words {pp/2m} is a constant vector in p space and {V(x)} in x space. Mathematically Shannon’s entropy Sp = - Sum over p Pe(p) ln(Pe(p)) and Sx =- Sum over x Pe(x) ln(Pe(x)) also yield constant vectors in their respective spaces namely ln(Pe(p)) and ln(Pe(x)) which are called information. In particular dSp = - Sum over p ln(Pe(p)) dPe(p) then ln(Pe(p)) behaves like a constant vector within a region of small dPe(p) changes. Note: Sum over p dPe(p)=0. One may consider a set of single valued energy levels ei which are very close (imagine a particle in a box with infinite walls) as a region in which dPe(p) changes (or dPe(x) ones) are very small. In this region, ln(Pe(p)) and ln(Pe(x)), information, represents a constant vector statistically defining the small region of ei states. This we argue is the statistical significance of information ln(P). One may note that both de ((1)) adn dSp and dSx change linearly with dPe(p) and dPe(x), but the vectors {pp/2m} and {V(x)} over p and x space and ln(Pe(p)) and ln(Pe(x)) over the same space are constant vectors, but not equivalent except for the ground state of the quantum oscillator for which they become very similar because Pe(p)=exp(-app) and Pe(x)=exp(-xx/a) with a being proportional to 1/ e which plays the same role as temperature in the classical ideal gas case. For other quantum bound systems, the two sets of vectors are different, but the information vectors seem to give a kind of statistical identity to small regions containing very close ei levels for which dPe(p) and dPe(x) are small. Thus, though one may argue for the presence of ln(P) in Shannon’s entropy S=-Sum over i P(i)ln(P(i)) in many ways, it seems that ln(P(i)) is a constant vector in a certain space i for small dP(i) and gives a statistical identity to that small region.
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.
Shannon’s entropy is defined for probabilities as - Sum over i P(i) ln(P(i)). For a probability density, such as spatial or momentum density which appear in quantum bound states i.e. P(x) = W*(x)W(x) or P(p) = a(p)* a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx), one defines: Sx = - Integral dx P(x)) ln{P(x)} and Sp = -Integral dp P(p) ln(P(p)) (1) and integrates over al space or momentum. For quantum bound states, however, one may integrate Shannon’s entropy using P(x) and/or P(p) only over the region of a single crest. Such a crest is of the form of an unnormalized probability density which has zero probability at the endpoints. Thus the total P(x) and/or P(p) might consist of a set of crests with possibly different heights. Thus instead of integrating over all x one may consider integrating only over the base of a particular crest to find a local entropy density so to speak. A reason for doing this is that the overall entropy may not satisfy dE = TdS, but an entropy for a crest might. We investigate this for an infinite well and an oscillator. For example, for a infinite potential well Sx over the full range 0 to L is independent of n, but by adding energy one creates pockets (crest entropies) which are very low as one has very high resolution.Thus we argue that a quantum bound system has an overall entropy, but also entropies for each crest (whether in x or p space). The latter may be linked to thermodynamics in a different manner than the former.