Shannon’s entropy is calculated using probabilities P(i) i.e. S= - Sum over i P(i) ln(P(i)). A probability is related to an uncertainty associated with event i, but this event may not be a physical event, but rather a measurement. In other words, measurements in a system may be made according to one distribution or another. This distribution may be converted into something which looks like a probability even though the system is completely deterministic. The entropy then calculated is based on the measurement distribution and not the physical system. As a first example, consider the completely deterministic motion of a bound classical system between two turning points. Person A may send values of x (one dimensional space) to person B, based on a certain distribution instead of sending the values based on time i.e. x(t1),x(t2) etc. One distribution which may be used is the uniform one i.e. P(x)= dx/L. Another distribution, used in the literature (1), is based on the time spent in each constant dx i.e. P(x) = Cdx/v(x) where v(x) is velocity. These two distributions lead to different Shannon’s entropies, but the underlying system is completely deterministic which begs the question: Why does one associate an entropy with it? The entropy seems to be associated with the sending of information. We try to apply these ideas to a quantum bound state. In the literature (2), one sometimes sees a Shannon’s entropy based on W(x)W(x) = P(x), where W(x) is the single particle bound wavefunction. This is a distribution, but is it based on uncertainty in the system? W(x)W(x) often consists of a set of crests of different heights and widths. We argue that the inherent uncertainty in x is due to the width of a crest. In other words each crest may be normalized to 1 and a Shannon’s entropy for each crest calculated. One must then weight the different crests. If one calculates the classical time associated with the base length of crest, this could be used as a weight which in the high energy level limit is associated with P(x)=Cdx/v(x) from (1), but it seems there is no reason to base the weight on time. The uncertainty in x is due to a superposition of exp(ipx)s which are not linked to time. Alternatively, one may use an unnormalized weight of the length of the base as uncertainty is only in x. One may then add a Shannon’s momentum entropy using a(p)a(p)=P(p) where W(x)=Sum over p a(p)exp(ipx).
Shannon’s entropy is calculated using probabilities P(i) i.e. S= - Sum over i P(i) ln(P(i)). A probability is related to an uncertainty associated with event i, but this event may not be a physical event, but rather a measurement. In other words, measurements in a system may be made according to one distribution or another. This distribution may be converted into something which looks like a probability even though the system is completely deterministic. The entropy then calculated is based on the measurement distribution and not the physical system. As a first example, consider the completely deterministic motion of a bound classical system between two turning points. Person A may send values of x (one dimensional space) to person B, based on a certain distribution instead of sending the values based on time i.e. x(t1),x(t2) etc. One distribution which may be used is the uniform one i.e. P(x)= dx/L. Another distribution, used in the literature (1), is based on the time spent in each constant dx i.e. P(x) = Cdx/v(x) where v(x) is velocity. These two distributions lead to different Shannon’s entropies, but the underlying system is completely deterministic which begs the question: Why does one associate an entropy with it? The entropy seems to be associated with the sending of information. We try to apply these ideas to a quantum bound state. In the literature (2), one sometimes sees a Shannon’s entropy based on W(x)W(x) = P(x), where W(x) is the single particle bound wavefunction. This is a distribution, but is it based on uncertainty in the system? W(x)W(x) often consists of a set of crests of different heights and widths. We argue that the inherent uncertainty in x is due to the width of a crest. In other words each crest may be normalized to 1 and a Shannon’s entropy for each crest calculated. One must then weight the different crests. If one calculates the classical time associated with the base length of crest, this could be used as a weight which in the high energy level limit is associated with P(x)=Cdx/v(x) from (1), but it seems there is no reason to base the weight on time. The uncertainty in x is due to a superposition of exp(ipx)s which are not linked to time. Alternatively, one may use an unnormalized weight of the length of the base as uncertainty is only in x. One may then add a Shannon’s momentum entropy using a(p)a(p)=P(p) where W(x)=Sum over p a(p)exp(ipx).
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.
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.
Given a coin, one has two complementary pieces of information and a corresponding probability of .5 and .5. Similar considerations apply to a die. As a result, one may get the impression that there exists a unique probability distribution in a probabilistic system and that this single probability scheme is linked to the constrained maximization of Shannon’s entropy or Fisher’s information. We argue however, that there may be different probability schemes in the same problem associated with different pieces of information which are relevant to the physics which occurs in the system. Thus, we argue that it is the physics which selects the probability distribution to be examined. For example, one may loosely think in terms of a detailed probability scheme versus a kind-of coarse grained one in the same system. For example, for a quantum free particle, P(x)=1 i.e there is the same probability for a particle to be at any x point. This, however, is based on velocity, i.e. the relationship between dx and dt. There is another piece of information in the system, namely mo which together with v yields momentum (either relativistic or nonrelativistic) .This need not be measured using a clock (which velocity requires), but by the impact of an impulse on a target. Thus, nonrelativistically p=m1v1=m2v2 give the same impulse, but have different velocities. Quantum mechanics shows that a free particle is governed by exp(ipx) which contains two unnormalized probability distributions cos(px) and sin(px). These are key for impulse interactions (i.e. the physics of interactions such as with a 2-slit apparatus etc), yet exp(ipx)/sqrt(L) maps into the “coarse-grained” P(x)=exp(ipx)exp(-ipx) / L = constant. P(x) is one probability scheme in the problem and exp(ipx) another (for a free particle). Each answers some physical questions, but one needs to know which probability to use (i.e. what information is relevant) to a particular situation. P(x) is relevant to motion in space not involving interactions. exp(ipx) is relevant to interactions (even with a magnetic vector potential causing no magnetic field) or with a 2-slit apparatus. Furthermore, in a quantum bound state, one has W*(x)W(x) = P(x) (W(x)=wavefunction =Sum over p a(p)exp(ipx)) and a*(p)a(p).=P(p). Each probability scheme corresponds to the analysis of a different kind of measurement, but various schemes may also be linked. Thus one may have to start with one to create the others. For example, exp(ipx) is the base for creating W(x)=Sum over p a(p)exp(ipx), which ultimately yields P(x)=W*(x)W(x). As a result, we argue that if probability exists in a system, it is not necessarily the case that there is a unique probability distribution. There may be various probability schemes linked with different information and different measurements/interactions. Thus, physics leads one to focus on a specific probability scheme. As a result, we argue that one must be careful when discussing pure statistical quantities and their extremization, because these quantities, like Shannon’s entropy and Fisher’ s information, contain a classical probability distribution, but they do not indicate which distribution to use in the case that a system has more than one. . As a result, the analysis of a system requires knowledge of the physics (interactions) occurring.We argue against thinking in terms of physics following directly from statistical considerations. The constrained extremization of one statistical quantity may yield a probability distribution related to one piece of information, but not to another. Furthermore, Shannon’s entropy and Fisher’s information pertain to classical probability 0<=P(x)<1 and Integral P(x)dx =1. The wavefunction, exp(ipx)/sqrt(L), is comprised of two probability distributions and ADD and MULTIPLIES statistically (i.e. OR and AND ) situations, but is involved in relative average calculations because it is not normalized. It is also different because it allows for positive and negative values. These are properties which do not follow from statistics using a classical P(x).
In the literature, Shannon’s entropy with spatial density as probability i.e. density(x) ln(density(x)) appears frequently. It seems, however, that this calculation of entropy assumes no knowledge of momentum, yet density=W(x)W(x) (for a bound state) where W(x) = Sum over p a(p) exp(ipx). In other words, the wavefunction W(x) contains information about the momentum distribution associated with P(p/x)= a(p) exp(ipx)/W(x). In classical statistical mechanics, entropy is calculated using exp[ - (p*p/2m + V(x))/T) which includes both momentum and spatial considerations. In a previous note (1), we calculated a possible expression for entropy using P(p/x)P(x). This includes Shannon’s spatial entropy and other terms even if one sums over p after forming Shannon’s entropy with P(p/x)P(x). In this note, we try to consider the interpretation of these two forms of entropy.
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 this note, we examine the behaviour of Shannon’s entropy density for some examples in quantum mechanics. We start with a quasiparticle probability P(x and p) = W(x) fp exp(ipx) (where W(x) is the wavefunction given by Sum over p fp exp(ipx)) and obtain a Shannon’s entropy (after summing on p) : Total entropy density = -P(x) ln(P(x)) - P(p) ln(P(p)) - W(x) x d/dx W(x) ((1)) It may be noted that if f(p)=f(-p) ((1)) may also be written as: Total entropy density = -P(x) ln(P(x)) - P(p) ln(P(p)) - fp p d/dp fp We consider the cases of a particle in a box with an infinite potential at the walls and the harmonic oscillator ground state. Finally we consider the influence of u(x)= [d/dx W(x)]/W(x) on the spatial change of Shannon’s spatial entropy as W(x) and u(x) are often periodic and this leads to a periodic entropy density. In general, entropy is obtained by integrating the entropy density and one may question whether one should examine the density at all. In (1), however, entropy density is considered and it seems that there are some interesting properties of Shannon’s entropy density in quantum systems. Furthermore, there has been some recent work in the literature which makes use of entropy density to create an extra potential term for the Schrodinger equation in cases of temperature.