Entropy is usually associated with probabilities (e.g. Shannon’s entropy= S = - Sum over i Probability(i) ln{Probability(i)} which suggests there is some uncertainty in the problem, captured by S. Entropy, however, also seems to be associated with the evolution of a system in time. Historically, entropy was introduced through the first law of thermodynamics: dE = TdS - PdV, before there was any formula involving probabilities. Furthermore, a constant entropy, which applies to an adiabatic process, means the process is reversible in time. In this note, we consider the case of a photon which is represented by a quantum wavefunction exp(-iEt+ip dot r), but is also reversible in the case of reflection/refraction, even from a moving interface. The wavefunction, we argue, is associated with probability in space-time and so there should be associated entropies (as we have argued in (1)), but at the same entropy should be constant as the motion is reversible. In (1), we considered Shannon’s entropy for t and x and used A1 |cos(Et)| and A2 |cos(px)| over half a cycle as probabilities. We introduce a third entropy associated with time, but not E in this note. Here, we focus on the case of refraction. For a stationary interface, E is conserved, but p changes. Thus there is a change in the uncertainty in x, but not in t and the process is reversible, with time S being constant.( The first law suggests that S=constant, which is keeping with time entropy, but not with spatial (x) entropy.) We suggest that there may be multiple types of entropy associated with different observables. We next consider the case of reflection/refraction from a moving mirror/interface. In such a case, neither E nor p is conserved for the photon. Both time and spatial (x) entropy change, but the process is reversible and entropy should be constant. We suggest there exists an entropy which is constant, but is based on: -Et+ p dot r = t |p| (-E/|p| + unit vector dot r/t) where r/t vector is minus the velocity of the moving interface and the unit vector points in the direction of momentum p. We consider an associated entropy and the first law of thermodynamics.
Entropy is usually associated with probabilities (e.g. Shannon’s entropy= S = - Sum over i Probability(i) ln{Probability(i)} which suggests there is some uncertainty in the problem, captured by S. Entropy, however, also seems to be associated with the evolution of a system in time. Historically, entropy was introduced through the first law of thermodynamics: dE = TdS - PdV, before there was any formula involving probabilities. Furthermore, a constant entropy, which applies to an adiabatic process, means the process is reversible in time. In this note, we consider the case of a photon which is represented by a quantum wavefunction exp(-iEt+ip dot r), but is also reversible in the case of reflection/refraction, even from a moving interface. The wavefunction, we argue, is associated with probability in space-time and so there should be associated entropies (as we have argued in (1)), but at the same entropy should be constant as the motion is reversible. In (1), we considered Shannon’s entropy for t and x and used A1 |cos(Et)| and A2 |cos(px)| over half a cycle as probabilities. We introduce a third entropy associated with time, but not E in this note. Here, we focus on the case of refraction. For a stationary interface, E is conserved, but p changes. Thus there is a change in the uncertainty in x, but not in t and the process is reversible, with time S being constant.( The first law suggests that S=constant, which is keeping with time entropy, but not with spatial (x) entropy.) We suggest that there may be multiple types of entropy associated with different observables. We next consider the case of reflection/refraction from a moving mirror/interface. In such a case, neither E nor p is conserved for the photon. Both time and spatial (x) entropy change, but the process is reversible and entropy should be constant. We suggest there exists an entropy which is constant, but is based on: -Et+ p dot r = t |p| (-E/|p| + unit vector dot r/t) where r/t vector is minus the velocity of the moving interface and the unit vector points in the direction of momentum p. We consider an associated entropy and the first law of thermodynamics.
In Part 1, we argued that it is no coincidence that the photon free particle wavefunction exp(-iEt+ipx), which we call a complex probability that conserves energy and momentum, has a real part which matches the electric field of a photon. In particular, we argued that even if one did not know Maxwell’s electromagnetic laws, one would postulate a force linked to a photon which behaved as cos(-Et+px). Here, we reexamine Maxwell’s electromagnetic equations. These are presumably deterministic differential equations and the photon solution with perpendicular electric and magnetic fields in a plane perpendicular to motion with each field changing as cos(-Et+px) for motion in the x direction, a seemingly deterministic solution. This deterministic solution, however, seems to be problematic for the following reason. If one considers one-dimensional reflection-refraction according to the prescription of (1), one writes a continuity equation for the electric field of an incoming photon with a reflected and refracted one, and another continuity equation for the first derivative. Physically, this appears to be a steady stream approach as well as a mathematical continuity one. The solution in (1) is obtained without any notion of probability until the last line: 1= Probability(reflect)+Probability(refract). The situation, however, is completely probabilistic from the beginning we argue and so one should understand how a deterministic electric (and magnetic field) is actually associated with probability. (Furthermore, photon interference and diffraction are also probabilistic.). We have argued in previous notes that examination of special relativity, in particular, the Lorentz invariant -Et+px = constant(x,t), leads to a probability for interaction of exp(-iEt+ipx) for both a particle with rest mass and a photon. This probability conserves E,p, but at first glance does display the inherent force in the problem. It is the inherent force which delivers E,p. Probabilistically, there is an uncertainty associated with the interaction time of hbar/E and with interaction position, hbar/p. These should be associated with the inherent force. As a result, the photon electric field, which acts as a force, cannot be precisely as described by cos(-Et+px) because this violates the notion of hbar/p and hbar/E. Rather, this formula is an expression of probability of the action of the force. One ultimately does not have a deterministic situation, we argue. In Part 1, we noted that a rest mass mo is associated with a force (inertia) as well as energy and momentum, which are proportional to mo. A photon has E and p and so should be associated with a force as it may render its energy to an object. The interesting thing about the photon is that it has no rest frame, even for the force. Given that exp(-imocc t) describes the probability associated with a rest force mo, we suggest that for a force linked with a moving photon, one should consider cos(-Et+px). This was already proposed in Part 1. This means that instead of considering the momentum as delivering the force, as it is proportional to mo for a particle with rest mass, for a photon, the electric field represents probabilistic force rendering, even though it is real and appears to be deterministic. Given that the electric field (inherent force) changes as cos(-Et+px), one may see that this force may change sign, but momentum must be in one direction. This suggests the presence of a second vector, together with the electric field in a quadratic arrangement with both becoming negative at the same time, i.e. both following cos(-Et+px). One may then create energy and momentum using Electric field dot Electric field and second field dot second field and electric field x second field for momentum. In such a case, energy and momentum would be cycle averages as the electric field changes as cos(-Et+px).
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.
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.
We suggest that quantum mechanics is concerned with preserving two conservations, one related to a kind of probability and the other to a mechanical balance.These two conservations exist at the level of a single particle and are interlinked. Specifically, the derivative d/dx, which measures the effect of a change in a kind of probability, also measures the associated resistance or impulse for a free particle. For a free particle, the “kind of probability” is exp(ipx) and d/dx exp(ipx) = ip exp(ipx) showing that there is an impulse related to p linked with the probability. In other words, a free particle moves through space in a probabilistic manner related to the resistance associated with the impulse p. These two conservations, a kind of probability and a kind of resistance, lead to two independent equations when one considers continuity at points which change momentum. We consider again the problem of a photon moving in the x direction which reflects or refracts at an interface along the y axis between n1-n2 (indices of refraction). Classically, probability is conserved i.e. 1= P(reflect) + P(refract) ((1)). The steady state problem, however, has the appearance of an equilibrium and should have a second conservation, namely one in space, i.e. a balance of pressure. This balance, linear in momentum, however, is the identical equation as ((1)). Thus, there are still two unknowns P(reflect), P(refract) and one equation. In the quantum case, double conservation may classically be manifested by the same equation. This seems to suggest, we argue, that ((1)) is a consequence of an equation based on p (which is linked to momentum). We try to find this association in two ways. First, we argue that the interface may exist at xo and that there are various momenta in the problem, We argue these represent pertinent information which does not appear in ((1)). We suggest that ((1)) seems to be a kind of magnitude equation with p and xo information being part of the direction of vectors i.e. p and xo may exist in a phase exp(i f(p,x)). Together with the conservation of pressure condition, we argue that exp(i f(p,x)) = exp(ipx). This argument, however, implicity involves consider the phase function exp(if(p,x)) as a kind of probability. Alternatively, we consider finding this probability follows from maximizing Shannon’s entropy subject to the periodic constraint i p x P(x), but we investigate why Shannon’s entropy form should be used, i.e. what is its physical significance?
In classical statistical mechanics, e.g. a Maxwell-Boltzmann (MB) gas, one tries to find the probability distribution of a particular quantity such as particle energy pp/2m. Thus there is a distribution in a quantity, making it the stochastic variable. This quantity, however, is subject to a conservation law i.e. elastic two particle scattering. In a previous note (1), we considered the case of reflection/refraction and called momentum the stochastic variable and noted that it too was linked with conservation. In fact, we argued that both the MB gas and reflection/refraction followed the same stochastic law: P(variable1) P(variable 2) = P(variable 1 + variable 2) whether variable 1 is a scalar or vector. In (2), however, we noted that exp(ipx), a free particle wavefunction, treats p as a parameter with x being the “probability” variable. This seems a little unusual because p, which is normally a variable in an MB gas and even in a quantum bound state i.e. a*(p)a(p) where W(x)=wavefunction = Sum over p a(p) exp(ipx) takes on the role of a parameter i.e. p=hbar/wavelength. Why this duality? Why is p a parameter for “x” of the probability distribution and yet at the same time a variable associated with conservation of momentum? In this note, we argue that unlike the distribution of a defining property of a particle such as momentum or energy, one may have a distribution associated with probability itself. This we argue is the case for a reflection/refraction problem which applies to both light and a particle. In the case of light, there is an interface of two media with different indices of refraction and in the case of a particle an interface with two constant potentials. In both cases, total energy of the photon/particle is constant, but there are two choices or probabilities, reflection or refraction, but they are linked with different times. A single photon or particle can only undergo one at a time. One knows exactly the two values of momentum, but the probability of each is unknown. Thus unlike an MB gas where energy/momentum is being redistributed because the two particles of the outcome exist at the same time, here a probability of 1 must somehow be distributed to the two events.. Given that the incident momentum is p, the reflected -p and the refracted p2, only considering a specific time event i.e. reflection or refraction, does not conserve momentum. In the MB case, a single time event i.e. scattering conserves both momentum and energy. Thus it seems that one does not deal with events in time. There is a distribution of probability in such a way as to ensure conservation of momentum, but does this probability already exist or get created at the interface point? We argued in (3) that if the scenario were deterministic one would have to follow it in time and treat each event separately. Rather we suggest that a distribution of probabilities allows for conservation of momentum, but this means that all three momenta exist at the interface x point as one is not considering an event at a precise point in time. This then allows for each momentum to have its own x distribution based on momentum p so that one may impose conservation of momentum and not have a deterministic time associated with the interface point.. This last idea of conservation of momentum appearing is not discussed in (3) and allows for p to behave like a parameter for x while at the same time being a variable which is conserved. In short, in an MB gas one distributes energy between two colliding particles in order to conserve total energy while for a quantum particle or photon which reflects/refracts one distributes probability to the two scenarios (reflection or refraction) such that momentum is conserved. Thus the nature of what is “distributed” is very different.