It is known that if a photon hits a surface with a different index of refraction (at an angle within a certain range), it has a specific probability “a” to reflect and 1-a to refract. In this note we ask: Where does this statistical nature come from? In previous notes, we have considered Fermat’s principle applied separately to explain two dimensional reflection and refraction. We have also used the idea of a difference of squares and energy conservation in one dimensional reflection/refraction to find values of the probabilities. The question we ask is: Why are there probabilities in the first place? Is there some general idea at work despite the complicated details of the photon interacting with the atoms of the new medium.We argue the general notion is due to the idea of spatial invariance which is linked to the translation operator d/dx which in turn in linked to a p impulse occurring at a given x .Thus the variables p and x are linked through the notion of an impulse existing at some x. The idea that d/dx may operate on a function of constant p and x to yield p, implies the function of p,x should have physical meaning. This is a problem as classically a particle moving with constant p should not be associated with any function of x except a constant. (All x points are the “same”.) Thus we argue that a “probability” in x is established with a constantly moving particle (or even photon). The eigenfunction of d/dx (or rather -id/dx) which exhibits some kind of spatial invariance is exp(ipx). This contains two different x probability distributions cos(px) and sin(px). We noted that d/dx is associated with p due to an impulse hit at say an x point. Thus various exp(ipx)s with different weights should appear together in a physical problem with impulses such as reflection/refraction. The weights of the various exp(ip1 x), exp(ip2 x) etc become linked to probabilities associated with the state 1, 2 etc. Thus the origin of these probabilities (i.e. their specific values) follow from the probabilistic form exp(ipx) which follows from d/dx. (In (1) we have argued that the x distributions in exp(ipx) may be further linked to the force which “created” the p in the first place, with different p values having different distributions cos(px), sin(px)).
It is known that if a photon hits a surface with a different index of refraction (at an angle within a certain range), it has a specific probability “a” to reflect and 1-a to refract. In this note we ask: Where does this statistical nature come from? In previous notes, we have considered Fermat’s principle applied separately to explain two dimensional reflection and refraction. We have also used the idea of a difference of squares and energy conservation in one dimensional reflection/refraction to find values of the probabilities. The question we ask is: Why are there probabilities in the first place? Is there some general idea at work despite the complicated details of the photon interacting with the atoms of the new medium.We argue the general notion is due to the idea of spatial invariance which is linked to the translation operator d/dx which in turn in linked to a p impulse occurring at a given x .Thus the variables p and x are linked through the notion of an impulse existing at some x. The idea that d/dx may operate on a function of constant p and x to yield p, implies the function of p,x should have physical meaning. This is a problem as classically a particle moving with constant p should not be associated with any function of x except a constant. (All x points are the “same”.) Thus we argue that a “probability” in x is established with a constantly moving particle (or even photon). The eigenfunction of d/dx (or rather -id/dx) which exhibits some kind of spatial invariance is exp(ipx). This contains two different x probability distributions cos(px) and sin(px). We noted that d/dx is associated with p due to an impulse hit at say an x point. Thus various exp(ipx)s with different weights should appear together in a physical problem with impulses such as reflection/refraction. The weights of the various exp(ip1 x), exp(ip2 x) etc become linked to probabilities associated with the state 1, 2 etc. Thus the origin of these probabilities (i.e. their specific values) follow from the probabilistic form exp(ipx) which follows from d/dx. (In (1) we have argued that the x distributions in exp(ipx) may be further linked to the force which “created” the p in the first place, with different p values having different distributions cos(px), sin(px)).
Quantum mechanics is known for its “square root” probability i.e. the existence of a wavefunction W(x) such that P(x) = W*(x)W(x). This begs the question: Why is this the case? In Maxwell’s electromagnetic theory of the photon, a similar situation arises except that one has energy density instead of P(x) which consists of .5C1 Electric field* Electric field + .5c2 Magnetic field*Magnetic field. The “square root” object is a physical observable (Electric field, Magnetic field) which both behave as cos(-Et+px) and so, more accessible than W(x). Alternatively one may use Fermat’s least time principle for light to show that i two dimensional problems (reflection/refraction) d/dx (used in extremization) is really the operator of momentum in that direction i.e. a spatial invariant direction. Thus A=px such that dA/dx=p = x component of momentum which may be generalized to A=-Et+p dot r. (p.r) may be written as -i ln(exp(ip dot r) with exp(i p dot r) representing a “kind of probability”. This probability may be considered a square root probability because for uniform motion (light) all x points along a ray should have equal probability. Thus exp(i p dot r) maps into a constant which happens if one takes the modulus. Both of the above approaches (using continuity at x or a kind of conservation of probability and a momentum) may be used to solve the problem of an incident beam of light moving in the positive x direction hitting a medium and both reflecting and refracting. For example see (1). The goal is to find the two intensity values associated with the reflected and refracted beams assuming the incident intensity is known. In this note, we take a different approach to finding the two unknowns, one that does not use the idea of the electric field proportional to cos(-Et+px) or the probability exp(ipx). We wish to solve the problem using only the idea of information and stress the importance of beginning with energy conservation a priori unlike the two approaches listed above. We postulate that there are two sets of information in this problem. First there is the notion of Energy = p(momentum in a vacuum) n c/n where n is the index of refraction. Thus the incident and reflected light are associated with pc1,c1 and the refracted with pc2, c2 with energy being the same for all. Second there is the idea of conservation of energy i.e. if ten photons with energy E hit the surface and a10 reflect and (1-a)10 refract, then the sum of the refracted and reflected energy equals the initial energy. Energy is then equated with intensity through Intensity/c = energy (1). Using only these two pieces of information we argue one should be able to find the intensity of both the reflected and refracted ray. This leads to an immediate problem because there are two unknowns and only one equation, namely conservation of energy i.e. I(A)/c1 = I(B)/c1 + I(C)/c2 where A is the incident beam, B the reflected and C the refracted. Writing as: I(A)/c1 - I(B)/c1 = I(C)/c2 (i.e. a spatial instead of time grouping) suggests a possible solution might be to employ a difference of squares noting that 1/c = p/E with E being a constant in the problem. Thus one may reduce the problem to A+B=C and p1A-p1B=p2C. This is of the form of a new kind of “square root” type probability, but leads to two equations with two unknowns (A is assumed to be known) which allows the problem to be solved.
In part I we argued that Newton’s first law implies that constant momentum (i.e. momentum conservation) is associated with spatial invariance i.e. a lack of information in a certain direction. This is equivalent to stating there exists no force in that direction. We argued that one may apply the generator for translation in the spatial direction i.e. d/dx for x and apply it to a function f(x) such that df(x)/dx = (px). The simplest form is f(x)= (px)x. We also argued that it is precisely this function which appears in Fermat’s least time principle because time = distance/speed and one may multiply time by |p| = momentum magnitude. Given that momentum and displacement are parallel vectors for a light ray, p dot r = p distance = (px)x + (py)y for a two dimensional problem (e.g. reflection or refraction). “Distance d” is the key function to use because dd= xx + yy implies that for fixed y, d d/dx = x/d = sin(theta1) (angle measured from y axis) so p d d/dx = (px) which is a conserved quantity if x is invariant. Thus conservation of momentum for two rays is equivalent to extremization. Extremization, we argue, suggests there may be a built-in physical mechanism which allows a free particle to correct itself if perturbed slightly just like a Maxwell-Boltzmann gas which extremizes entropy subject to constant average energy does. This built-in mechanism should be linked to motion in space. Writing (px)x as ln(exp(ipx), one not only obtains an eigenvalue equation i.e. -d/dx exp(i (px) x) = (px) exp(i px x), but the notion of a relative change arises if exp(i px x) is interpreted as a kind of probability. A relative change is natural because an absolute one ignores the idea that probability may differ with x even though (px) is constant. Thus from the idea of invariance in space found in Newton’s first law, which is equivalent to conservation of momentum in that direction, the equation d/dx f(x) = (px) leads directly to -id/dx exp(i px x) = (px) exp(ipx x) where exp(i px x) represents a physical probability type of wave which stablilizes the constant (px) motion because this motion is equivalent to extremizing time or |p| time or p dot r / speed of light in Fermat’s principle. In other words, the quantum mechanics of a free particle is associated with stabilizing wave behaviour exp(ipx x) which was not seen experimentally at the time of Newton or Fermat, but seems to be part of their formalisms, with Fermat’s principle derivable from Newton’s first law.
In Newtonian mechanics a particle with constant momentum moves deterministically according to x=p/m t. Even a quantum object such as a photon or electron should satisfy this equation on average to give meaning to the notion of the speed of light in a vacuum, for example. Thus it seems surprising that a quantum free particle is also associated with exp(ipx) which only contains momentum and is periodic and complex. Furthermore the physical consequences of exp(ipx) are real as shown in a two-slit interference experiment etc. In this note, we attempt to discover why exp(ipx) exists in the first place by examining the example of the reflection/refraction of light which occurs in the macroscopic classical world. Basically our argument is based on analyzing two important pieces of information in the system, namely the scalar energy (or particle count) and the vector momentum. A main point we make is that energy or particle number considerations are linked to determinism i.e. a specific time of interaction. One may note that the interaction x is fixed by the two medium barrier. This means a single incident photon exists in one time range and the reflected OR refracted photon in another. It is possible to compare the incident photon to either the reflected or refracted or to a sum of probabilities. In particular, a single photon has an energy E and a count of 1. If it reflects, it has an energy E and count of 1 so there is a balance, likewise for refraction. Thinking in terms of probabilities “a” and 1-a to reflect/refract: 1= a + 1-a. Can one perform such a matching for momentum, which is a physical piece of information? We argue one cannot. For example, a single incident photon has a momentum p which is linked to the impulse it may deliver. If it refracts, “all of a sudden” it is linked with momentum of p2 = n2 p where n2 is the index of refraction in the second medium and n1=1. This seems problematic because a physical impulse of 2p is all of a sudden an impulse of 2p2. Thus there exists a big problem which seems to require the notion of probability and all three momentum values p, -p and p2 considered together, unlike the situation of energy/particle number for which a comparison of the incident photon and one possible result yields a balance. Combining all three momenta, however, is also problematic in a deterministic picture because the incident photon exists in one time range and the reflected or refracted in another. How can one possibly compare them together? The solution seems to be to argue that the incident photon is only deterministic on average. In particular, if it has an x probability distribution over a certain length (called a wavelength), one does not know the precise time at which it hits “x” the medium interface. In such a situation, there is time uncertainty which allows all three momenta to be compared at the same x. This, however, leads to a major consequence, namely that so-called deterministic motion associated with constant p is deterministic on average, but contains repeated pockets of x probability distributions such that time is not deterministic within. We try to examine these ideas in this note.
It was argued in (1) that Fermat’s least time principle for light (in a two dimensional (x,y) reflection or refraction scenario) minimizes time by actually varying it in terms of a spatially invariant variable i.e. x (i.e. taking d/dx) with the flat face of the mirror/medium along the x axis at y=0. This is equivalent to conserving momentum in the x direction. Thus d/dx is associated with (px) which may be associated with the Lorentz invariant -Et+p dot r. In other words, d/dx emerges as a momentum operator (up to a factor) and d/dt as an energy operator, yet both are generators in time and space linked with invariance of constant values i.e. conservation. Furthermore an eigenvalue equation: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt) automatically (by the nature of eigenvalue) leads to orthogonal eigenfunctions linked to conservation of energy and momentum. In special relativity rest mass mo represents a constant in time which may exist at any point x. In fact, it is often thought of as a point, for example a center of mass point. The actual object with mo, however, may be large with moving parts all within a “container”. What then does x,t in -Et+px mean for constant E,p? It seems that x,t are center-of-mass coordinates for the overall system. We argued in previous notes that 1/p is proportional to a wavelength so for p→0, mo may be at any x point with the same probability. Within the system with moving parts, however, there is motion and an x scale for each moving piece. For a single particle bound system, there is a single energy at each x which is like mo and a potential V(x) and KE(x). In (2) we argued that given the free particle relationship -EE+pp= -momo (c=1) one may retain a constant E value by replacing E with E-V(x) and pp with <pp> (and not ). How do the probabilities appear? They are linked to -Et+px with frequency hbar/E and wavelength hbar/p and are specifically manifested by exp(-iEt) and exp(ipx). For and <pp> we use spatial probabilities i.e. W(x) = Sum over p a(p)exp(ipx). Each exp(ipx) is linked to the operator -id/dx and invariance, but pp is linked with the operator -d/dx d/dx and one does not use an eigenfunction of this operator. Thus the notion of conservation of momentum is retained in the exp(ipx)s and their linear combination W(x) yields a function such that an average kinetic energy exists at each x point i.e KE(x)=KEclassical = -1/2m d/dx dW/dx / W. This is combined with a V(x) to ensure constant E at each x point i.e. create a kind of mo piece linked with id/dt. We argue that E being constant at each x is a statistical equilibrium type of result. Each exp(ipx) is associated with an average motion of x=p/m t and given that exp(ipx) and exp(-ipx) are included, there is also the notion of which is positive and negative in different regions and integrates to 0. prms(x) from <pp> becomes the classical p(x). Thus bound state quantum mechanics seems to be based on a free particle treatment which follows from the special relativistic Lorentz invariants -EE+pp=-momo and -Et+px = constant, directly linked to conservation ideas. Conservation of momentum of an internal bound particle is maintained through exp(ipx), while <pp> by itself is not linked with conservation as there is no eigenstate, but is combined with a V(x) to create a constant En linked with the entire system (and P=0) which then becomes associated with invariance in time and the generator id/dt. In this note, we try to argue that the notion of W*(x)W(x) (where W(x)=wavefunction= Sum over p a(p)exp(ipx)) is spatial density and P(p)=a*(p)a(p) based on spatial invariance and conservation of momentum for a single particle bound state even though there seems to be no spatial invariance or notion of conservation of momentum in W(x) which is simply a solution of the time-independent Schrodinger equation. One may move from a W(x) scenario (i.e. Schrodinger equation) to a W*(x)W(x) picture, by trying to “pick out” invariant (p only terms) using the orthogonality of the exp(ipx)s. Thus W*(x)W(x) (spatial density) emerges naturally in this approach.
In a previous note (1) we argued that quantum square root probabilities follow from the consideration of light reflecting and refracting at a surface along the y direction at x=0. No knowledge of Maxwell’s periodic photon electric field is needed. The photon number and energy are conserved at a single photon level, but flux/intensity may be included through (2) AA/c1 = BB/c1 +CC/c2 where AA is the intensity of the incident beam (AA>0), BB the reflected and CC the refracted. There are two unknowns B,C if A is known and one equation which may be broken into two A+B=C and A/c1 -B/c1 = C/c2. For n1<n2 (n=index of refraction B<0). Thus we argue that there is an underlying square root probability. In this note we first heuristically suggest the form for the square root probability. A key point is that “particle” features such as E and p (for a photon) govern the dynamical wavelike probability. This also applies to a particle with rest mass. We also ask why this square root probability has not appeared in dynamical theories and argue that it appears in special relativity (i.e. a Lorentz boost) for a particle with rest mass mo at x=0 at t=to. As seen from a frame moving at constant speed -v, this particle moves with speed v. Thus x’=vt’, yet a 2x2 Lorentz transformation is applied to (x=0,t=to). Only one relation is needed because x’=vt’ and p’=E’/cc v suffice to yield the variables x’ and p’. Thus we argue that a Lorentz transformation describes more than is necessary for a simple particle. The 2x2 nature of the problem we suggest describes more than a simple particle. It leads to the Lorentz invariant -Et+px ,but x=vt so why should x and t appear as if independent? We argue that -Et+px=A is compatible with dA/dx |constant t = p and dA/dt |constant x = -E. P and E are “particle/photon” properties, but exp(ipx) or exp(-iEt) represent eigensolutions of -id/dx and id/dt. Thus if x and t are independent exp(ipx) may be interpreted as existing at all x points (as there is no time). This is the picture of an unusual probability which is two dimensional, with components which may be positive or negative as in the reflection/refraction problem. Thus we argue that the notion of a wave probability or square root probability not only exists in the reflection/refraction problem, but in the case of special relativity with what looks like an unnecessary 2x2 matrix from the point of view a particle (mo, x=0, t=to) describing an associated square root wave like probability completely dependent on the particle properties E and p.
In a previous note (1) we argued that the translational generator d/dx being associated with momentum (px) already appears in Fermat’s principle for both reflection and refraction. We suggested that d/dx is linked to an invariant spatial variable which in turn is associated with momentum conservation in the same direction. This may be written as: d/dx (px)x = (px). Alternatively (px)x = -i ln(exp(i px x) ) with exp(i px x) being a directional probability distribution (i.e. values for p and -p differ) and ln(exp(i px x)) being of the form of information in information theory. One may then ask: How does one link average momentum in the case of a bound state with a potential V(x)? In general, such a bound system is solved using the time-independent Schrodinger equation i.e. conservation of energy using d/dx twice and a p(rms) not a p average obtain from -id/dx ln(W) = p(x) where W(x) is the wavefunction and pave= W dW/dx. After all, the form exp(i (px) x) may be obtained from considerations of d/dx and p, not energy conservation (even though E=pc for light). We argue that exp(ipx) is a directional probability. If both p and -p are present then: p exp(ipx) -p exp(-ipx) is not zero because the spatial probability distinguishes between the two. Only by multiplying exp(ipx) and exp(-ipx) does one remove motion. If p and -p are present, however, as in an OR situation, i.e. exp(ipx) + exp(-ipx), then it is as if the particle is bound or localized and P(p,x)+P(-p,x) = C cos(px). We consider the two components of exp(ipx) in: Integral dx cos(px) sin(px). Why does this equal 0 from a momentum perspective? sin(px) = 1/p d/dx cos(px). Thus the integral is: Integral dx { P(p)+P(-p)} momentum(P(p)+P(-p)) } or the average momentum at x. Overall the particle is bound and so there can be no momentum globally so the integral must be zero and hence cos(px) and sin(px) must be orthogonal over a period even though they represent the same p value. This same idea holds for a linear combination of {P(p,x)+P(-p,x)} sums for different p’s. (We consider overall positive or negative parity.) Thus: Integral dx W dW/dx = 0 holds for even-odd reasons, but also for the physical overall zero momentum. Even with a potential present: -i WW d/dx ln(W) = Pave(x). In other words, bound state quantum mechanics is linked to W dW/dx integrating to 0 even though there are positive and negative pockets at various x values. This follows from the orthogonality of cos(px) and sin(px) over a wavelength from a free particle which conserves momentum. This result holds for any V(x) even though V(x) fixes the coefficients of the linear combination of Sum over p a(p) {P(p,x)+P(-p,x)} using energy considerations. Zero average momentum follows from momentum conservation found in exp(i (px) x). In fact we have argued that the unusual probability wave form exp(i (px) x) exists to ensure constant momentum (momentum conservation) for a free particle. We argue that considering average momentum equal to 0 overall leads to ideas of zero point energy, orthogonal energy states and a crest/trough pattern without considering the time-independent Schrodinger equation which is based on energy conservation. Thus we argue that ideas related to d/dx and momentum carry over from Fermat’s principle into a bound quantum state and its average momentum which is often not considered.