Quantum free particles are represented by exp(ipx) which we argue is a dynamical probability which maps into a static probability exp(-ipx)exp(ipx) i.e. the usual spatial density. Shannon’s entropy = - Sum over i P(i) ln(P(i)) is often associated with the static probability P(x)=W*(x)W(x) (1), but in (2) we argue that one may introduce a different probability C1 Real(W(x)). In this note we consider both in different examples. In particular we argue that the two different probability definitions are ultimately linked with momentum conservation. The form W*(x)W(x) introduces local nonconstant spatial densities if there are mixed exp(ip1x) and exp(ip2x)’s within W(x). If there are not, this form W*(x)W(x) does not distinguish space i.e. exp(ipx)exp(ipx+ib) is still a constant. Nevertheless interference may still occur which may be linked with entropy. Thus we first examine the case of exp(ipx) + exp(ipx + ib) where b is a constant phase shift. W*(x)W(x)=constant which does not lead to a structure different from exp(-ipx)exp(ipx), yet W(x)=C2{exp(ipx) + exp(ipx+ib)} interfere. On the other hand |cos(px)| and | cos(px)(1+cos(b)) - sin(px)sin(b) | are not the same distributions (one is a parity eigenstate, the other is not) and so we argue there is a different local entropy density. Thus introducing extra information i.e. b changes local density and entropy density if one adjusts the definition of probability used. We next consider W(x)= exp(ip1x)+exp(ip2x). One may also use the probability C3 Real|W(x)|, but here we are interested in an association with conservation of momentum. For a given exp(ipx), exp(-ipx)exp(ipx)=1. We argue that this represents a steady state picture i.e. a kind of equilibrium in which all x points are treated the same. W*(x)W(x) breaks this equilibrium scenario by introducing a term 2 cos( (p1-p2)x ). Thus both local spatial density and local entropy (based on W*W) are different and do not represent the “equilibrium” scenario. We further argue that momentum conservation may be linked with this equilibrium type of density and entropy density at least in the case of the transition amplitude <p1 V(x) p2>. V(x)exp(ipx) yields a sum of exp(ipx)’s, but only exp(ip1x) survives the integral over dx. The others create spatial densities which are not constant and so represent local entropy densities which do not treat all x points the same i.e .is not an equilibrium one. Thus we try to use Shannon’s entropy with two different probability definitions to analyse different quantum interference scenarios and argue that momentum conservation (and possibly others) may be linked with local entropy density in some situations.
Quantum free particles are represented by exp(ipx) which we argue is a dynamical probability which maps into a static probability exp(-ipx)exp(ipx) i.e. the usual spatial density. Shannon’s entropy = - Sum over i P(i) ln(P(i)) is often associated with the static probability P(x)=W*(x)W(x) (1), but in (2) we argue that one may introduce a different probability C1 Real(W(x)). In this note we consider both in different examples. In particular we argue that the two different probability definitions are ultimately linked with momentum conservation. The form W*(x)W(x) introduces local nonconstant spatial densities if there are mixed exp(ip1x) and exp(ip2x)’s within W(x). If there are not, this form W*(x)W(x) does not distinguish space i.e. exp(ipx)exp(ipx+ib) is still a constant. Nevertheless interference may still occur which may be linked with entropy. Thus we first examine the case of exp(ipx) + exp(ipx + ib) where b is a constant phase shift. W*(x)W(x)=constant which does not lead to a structure different from exp(-ipx)exp(ipx), yet W(x)=C2{exp(ipx) + exp(ipx+ib)} interfere. On the other hand |cos(px)| and | cos(px)(1+cos(b)) - sin(px)sin(b) | are not the same distributions (one is a parity eigenstate, the other is not) and so we argue there is a different local entropy density. Thus introducing extra information i.e. b changes local density and entropy density if one adjusts the definition of probability used. We next consider W(x)= exp(ip1x)+exp(ip2x). One may also use the probability C3 Real|W(x)|, but here we are interested in an association with conservation of momentum. For a given exp(ipx), exp(-ipx)exp(ipx)=1. We argue that this represents a steady state picture i.e. a kind of equilibrium in which all x points are treated the same. W*(x)W(x) breaks this equilibrium scenario by introducing a term 2 cos( (p1-p2)x ). Thus both local spatial density and local entropy (based on W*W) are different and do not represent the “equilibrium” scenario. We further argue that momentum conservation may be linked with this equilibrium type of density and entropy density at least in the case of the transition amplitude <p1 V(x) p2>. V(x)exp(ipx) yields a sum of exp(ipx)’s, but only exp(ip1x) survives the integral over dx. The others create spatial densities which are not constant and so represent local entropy densities which do not treat all x points the same i.e .is not an equilibrium one. Thus we try to use Shannon’s entropy with two different probability definitions to analyse different quantum interference scenarios and argue that momentum conservation (and possibly others) may be linked with local entropy density in some situations.
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 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.
Quantum mechanics is based on two different probability schemes, namely W(x) =wavefunction and W*(x)W*(x)=P(x), with one being linked to the other. Why should there be two probability schemes? In (0) we argued that there are two spatial behaviour schemes because an impulse p (momentum) is automatically associated with two spatial distributions cos(px) and sin(px) in exp(ipx), yet a particle with constant p moves as x=p/m t. Thus if one is only concerned with motion, each x point has the same weight, and this is physically measurable as a photon or electron travels a certain distance in a certain time in free space. On the other hand, an impulse interaction is associated with exp(ipx) is a completely different spatial picture, and this too is measurable as in a 2-slit experiment. Thus it seems the two schemes are measured differently. We argue that this is an important issue because if there are two x-probability schemes there should be different entropies associated with each. Ttraditionally (1), one sees two quantum entropies: - W*(x)W(x) ln {W*(x)W(x)} and - a*(p)a(p) ln{ a*(p)a(p) } where W(x) = Sum over p a(p)exp(ipx). These are associated with the x=p/m t scheme or x-dependence based on motion or average motion. W*(x)W(x) represents spatial density which should be measured presumably by some sort of scattering, but we ask whether some information is lost in this approach? In particular, we note that a wavefunction may be obtained for a bound state. In the high energy limit, the envelope over the density W*(x)W(x) is said (2) to match C/v(x) from classical physics where .5mv(x)v(x) + V(x) = E. C/v(x), however, is proportional to delta t (with delta x being constant) and has nothing to do with the strength of an impulse hit. In the classical case, there is a single v(x) and so a single p(x) in each delta x thus no impulse entropy within a delta x i.e. at a point x. In quantum mechanics, however, we argue that a constant p automatically has an x probability distribution exp(ipx) associated with it and there might be an entropy linked to this.This entropy is associated with a probability to receive a p impulse at x and is so not linked to interference while the W*(x)W(x) is. In (3) we introduced the impulse or wavelength entropy (for a free particle |cos(px)|/C ln{| cos(px)|/C},, but here we wish to investigate its properties more closely.
In the literature, one often finds a quantum expression for Shannon’s entropy of the form: - (W*W) ln(W*W) ((1)) where W*W is spatial density i.e. W(x)= the wavefunction. This implies W*(x)W(x) is an independent probability. Given that W(x)= Sum over p a(p) exp(ipx), and W*W= Sum over p1, p2 a(p1)a(p2) exp(ip1 x) exp(ip2 x), one may see that for p1/-p1 in the latter with a(p)=a(-p) leads to cos(p1x) terms. These are positive and negative in different x regions. In fact, in many cases, the overall spatial density exhibits peaks and valleys suggesting that W*(x)W(x) is correlated, i.e. not independent we argue. Furthermore, the form ((1)) uses only position when there is an expectation value for < p*p> an , both functions of x, where p is momentum. Thus, the variance of p-ensemble average may change with x, yet position only is used in the computation of Shannon’s entropy, which we argue seems unusual. In a previous note (1), we suggest using a(1)exp(ip1 x) a(p2)exp(ip2 x) as the probability for Shannon’s entropy. This leads to a spatial form 2W x d/dx W which matches ((1)) for the ground state of the quantum oscillator case. For the ground state oscillator, W*W=C exp(-ax*x) i.e. a Gaussian. This suggests that for this case only, W*(x)W(x) represents an independent probability which may be used in Shannon’s entropy, we argue. We attempt to consider these issues in this note.
In previous notes, we have argued that one may obtain exp(ipx) (the spatial free particle probability) from the maximization of P1(x) ln(P1(x)) + i px P1(x), and a similar expression for P2(t). This yields exp(-iEt+ipx). If one integrates i Integral px exp(ipx) dx from px=0 to 2*3.14/p, one obtains i 2*3.14/p as a spatial entropy and i2*3.14/E as a temporal one associated with the complex probability exp(-iEt+ipx). If one considers classical motion of a free particle in space, then there is no uncertainty in time or space and so one might create a classical level entropy of: -E dt + p dx, where dt=hbar/E and dx=hbar/p.This yields -Edt+pdx=0 and one might equate this with a classical entropy dS as done in (1). This overall classical entropy is different from the spatial and temporal wave entropies dx and dt. If one does not wish to work at the classical level, one might try to use the hbar/p, hbar/E entropies in a problem requiring exp(-iEt+ipx), such as 1-D reflection-refraction at an n1-n2 index of refraction junction (x=0). In such a case, it seems that one does not use entropies, but rather their reciprocals which indicate the amount of resolution information present, i.e. L/dx = Lp/hbar. As a result, conservation of E and p may be reformulated in terms of a balance of resolution information which is the reciprocal of dx, i..e p (in the spatial case). For example, a particle at rest may break into two pieces with p and -p. The resolution is proportional to p initial =0 and after one has p and -p (i.e. conservation) of momentum which may be argued to be equivalent to conservation of resolution information. Using the notion of a balance of the reciprocal of entropy (i.e. p which is linked to conservation of momentum), one might interpret: Ap - Bp = C p2 where p2=n2 p in terms of resolution and not momentum conservation, i.e. A-B = C p2/p where p2/p = n2 is the greater relative resolution of the p2 photon. The idea of both a wave entropy (both temporal hbar/E and hbar/p) and its reciprocal resolution information seem to arise as well an overall classical entropy -Edt+pdx = 0 if dt=hbar/E and hbar/hp = 0. dt=hbar/E and dx= hbar/p We have argued in previous notes that one may obtain: dt=hbar/E and dx=hbar/p ((1)) as independent dx and dt fluctuation from the Lorentz invariant. One may show that -Et+px is equivalent to the classical relativistic or nonrelativistic action Lt if Lt=.5movv t (nonrelativistic) or Lt = -mo sqrt(1-vv/cc) t (relatvistic) and x/t=v ((2)) The reason that one recasts Lt as -Et+px is to use relativistic variables. The notion of an action remaining unchanged under a fluctuation delta x(t) , i.e. at t one considers x1 instead to x=vt, leads to: 0 = -E delta t + p delta x = 0 which has a solution of ((1)). In (1), however, the notion of an overall entropy is introduced such that: dS = 0 = -E delta t + p delta x ((3)) Given that one really has a probabilistic scenario (free particle quantum mechanics) associated with ((1)), i.e. exp(-iEt+ipx) = probability which ensures ((1)) and also spatial and temporal invariance by the unit modulus (less information than exp(-iEt+ipx)), then one might consider the notion of an overall entropy.
In classical physics, a particle traveling to the right is the mirror image of one traveling to the left. In terms of a spatial density (the time spent in each constant dx), the value is the same, as is the energy. A similar notion of such a spatial invariance is in the classical conservation of energy equation for a bound particle: p(x)p(x)/2m + V(x) = E. Quantum mechanics seems to be different because exp(ipx) and exp(-ipx) are distinctly different spatial probabilities, i.e. they are dynamic. It is not just that case that a particle moves to the right for exp(ipx) and to the left for exp(-ipx). The impulse hit probabilities (represented by exp(ipx) and exp(-ipx)) are distinctly different. If one wishes to create an OR situation and write: W(x) = Sum over p a(p)exp(ipx) and use this in a conservation of energy equation, then simply including only positive or negative p values in W(x) leads to a complex average kinetic energy -1/2m d/dx dW/dx / W which cannot be used with V(x) and E. Furthermore, W_ (only negative p) and W+ (only positive p) values do not yield the same average kinetic energy. Fortunately, given that this is a statistical average for a localized object (somewhat analogous to a gas), one may combine W+ + W- and obtain a real kinetic energy. We have argued in previous notes that there exists a quantum-classical duality. For example, for a free particle one has two independent notions. One is x=vt which describes the center-of-mass of the particle. We take this to represent a state description which treats v and -v as mirror images. exp(ipx), on the other hand, represents probabilistic impulse hits and does not define the state by itself, we argue, only its interactions. Interactions are physical and define what happens in an interacting system. One may, however, measure a system, i.e interact with exp(ipx), and this should define a measurement state ( a second state ensemble average). For example, exp(ipx)exp(-ipx) = 1 which is equivalent to the constant dt in each dx associated with x=vt. Thus, the two states coincide, but exp(ipx)exp(-ipx) means that one may place the 2-slit apparatus at any x point and is not about the center-of-mass, as x=vt is. In some cases, the two states are quite different. W*(x)W(x) represents an interaction state, but vrms(x) from .5mvrms(x)vrms(x) = -1/2m d/dx dW/dx / W represents a center-of-mass state description. (Note: in these cases a state means an average state, i.e. an average over many ensemble entries.) If one takes W*(x)W(x) and integrates over x, then interference terms vanish and one obtains a classical counting of distinct p value states with their weight a*(p)a(p).This, we argue, leads to the notion of orthogonality of probabilities. In classical probability, there is no such notion, but there is the notion of AND (multiply) probabilities. Given exp(ipx) and exp(-ipx), if one multiplies one removes p, i.e. the interactions and only sees the state left. W*(x)W(x) yields a set of peaks which represent a pattern of probabilistic hits in space which define a state which one would measure in space. The interference pattern, however, must be one such that if one integrates over x, one only obtains a classical sum of probabilities a*(p)a(p)’s as mentioned above. This leads to the notion of orthogonal exp(ipx)’s (which follows mathematically anyway). If one considers different bound En states, then these two are orthogonal because W*n1(x)Wn2(x) mean n1 and n2 are associated with different energies and integrating over x cannot show a result which looks like a state result, where W*n(x)Wn(x) shows a state result. In order to achieve this orthogonality (linked directly to the notion of a state) one must have that Wn(2) usually has 0 points and is positive and negative with the possible exception of a ground state. In other words, if Wn1*(x)Wn2(x) integrated over x is not 0, one violates conservation of energy. As a result, we argue an average probability for a bound state must statistically include +p and -p contributions and must create a function which has positive and negative peaks (except possibly for the ground state), i.e. W(x) must have 0 points. This W(x) is linked to a measurement state W*(x)W(x), but this is strictly considering exp(ipx)s. We argue that there are two pieces of information for a free particle, x=vt and exp(ipx). There must also be an average state linked to x=vt and this given by KEave(x) = .5m vrms(x)vrms(x) + V(x) = E which shows that the particle may exist and have average kinetic energy even at W(x)=0 points. When measuring a system, one interacts with the exp(ipx)s and so the W*(x)W(x) state is the one which is traditionally presented, but whether one measures the center-of-mass motion or not, one also has the notion of the x=vt, i.e. vrms(x) state (average) as well, we argue.