Addendum Sept. 8, 2022: The conservation of photon momentum parallel to the moving mirror holds i.e. p(incident) sin(AA) = p(reflected) sin(BB). (This is the same approach that may be used to derive Snell's law, however, the angle BB is unknown.) If p=hbar/wavelength, then pi/pr = wavelength r / wavelength i and the momentum result matches the wavefront approach described in the note. An independent relation between cos(BB) and cos(AA) may be obtained using Fermat's principle i.e. cos(BB) ={ -2v/c+ (1+vv/cc)cos(AA) } / {1-2(v/c)cos(AA)+vv/cc} from Ref(2) (For example, one may take the parallel momentum expression above, write p=hbar/wavelength and insert the wavelength relationship given in Ref 2. Using the cos(BB) - cos(AA) given directly above one may see that the two approaches are compatible. The concept of impulse exists in Newtonian mechanics, but wavelength is not associated with momentum in such a case Even for light (which has a wavelength) reflection from a stationary mirror or refraction into a stationary medium, the idea of momentum without wavelength may be applied (1). In quantum mechanics, however, momentum is directly linked to wavelength through p=hbar/ wavelength, but what does this relationship mean for interactions? Wave mechanics were known long before quantum mechanics and special relativity and often involve following two wavefronts as they interact to see if their spacing remains the same. The interaction itself, however, seems to be very much like a Newtonian impulse one. Thus there are two types of “mechanics” occurring, one linked to impulse and the other to wave features i.e. delays which disrupt the spacing of wave fronts. In particular we try to associate this idea with the reflection of light from a moving mirror. In (2) the author considers such a scenario with two wavefronts, each undergoing identical impulse-like interactions, but calculates how the movement of the mirror during the time the two wavefronts react affects the distance between the two reflected wavefronts. The author argues that this approach allows one to obtain the same results as special relativity. In this note we argue that such an analysis may give insight to quantum mechanical interactions i.e. help clarify how momentum and wavelength are linked in a practical problem. As a result, we argue that quantum mechanics of a free particle and special relativity are closely linked and that quantum mechanics like wavefront mechanics combines Newtonian interactions with an uncertainty in the particle position i.e. two extreme points separated by a wavelength.
Addendum Sept. 8, 2022: The conservation of photon momentum parallel to the moving mirror holds i.e. p(incident) sin(AA) = p(reflected) sin(BB). (This is the same approach that may be used to derive Snell's law, however, the angle BB is unknown.) If p=hbar/wavelength, then pi/pr = wavelength r / wavelength i and the momentum result matches the wavefront approach described in the note. An independent relation between cos(BB) and cos(AA) may be obtained using Fermat's principle i.e. cos(BB) ={ -2v/c+ (1+vv/cc)cos(AA) } / {1-2(v/c)cos(AA)+vv/cc} from Ref(2) (For example, one may take the parallel momentum expression above, write p=hbar/wavelength and insert the wavelength relationship given in Ref 2. Using the cos(BB) - cos(AA) given directly above one may see that the two approaches are compatible. The concept of impulse exists in Newtonian mechanics, but wavelength is not associated with momentum in such a case Even for light (which has a wavelength) reflection from a stationary mirror or refraction into a stationary medium, the idea of momentum without wavelength may be applied (1). In quantum mechanics, however, momentum is directly linked to wavelength through p=hbar/ wavelength, but what does this relationship mean for interactions? Wave mechanics were known long before quantum mechanics and special relativity and often involve following two wavefronts as they interact to see if their spacing remains the same. The interaction itself, however, seems to be very much like a Newtonian impulse one. Thus there are two types of “mechanics” occurring, one linked to impulse and the other to wave features i.e. delays which disrupt the spacing of wave fronts. In particular we try to associate this idea with the reflection of light from a moving mirror. In (2) the author considers such a scenario with two wavefronts, each undergoing identical impulse-like interactions, but calculates how the movement of the mirror during the time the two wavefronts react affects the distance between the two reflected wavefronts. The author argues that this approach allows one to obtain the same results as special relativity. In this note we argue that such an analysis may give insight to quantum mechanical interactions i.e. help clarify how momentum and wavelength are linked in a practical problem. As a result, we argue that quantum mechanics of a free particle and special relativity are closely linked and that quantum mechanics like wavefront mechanics combines Newtonian interactions with an uncertainty in the particle position i.e. two extreme points separated by a wavelength.
According to (1) Fermat’s minimum time principle is identical to Hyugen’s wave principle. In a previous note (2) we considered two ray light systems (in particular Snell’s law and reflection from a mirror moving at constant velocity v) and argued that Fermat’s principle is equivalent to a hypothetical velocity approach where H sin(AA) = v(A relative) and H sin(BB = v(B relative). Here H is the hypothetical velocity which acts as a hypotenuse and AA and BB are the incident and reflected rays in the case of a moving mirror (measured relative to the normal) and the incident and refracted rays for Snell’s law. The hypothetical velocity approach, however, is exactly in the form needed for conservation of momentum parallel to the medium surface which holds for both the moving mirror and Snell’s law if: 1/wavelength(A) is proportional to 1/v(A relative) and the same result for B. This is equivalent to considering a fixed frequency f which applies to A and B (and is not 1/energy proportional to wavelength) so that vrel(A) = (same frequency) wavelength(A) and the same for B. As a result, conservation of momentum parallel to the medium surface is implicitly included in Fermat’s principle. In the case of quantum mechanics of a free particle represented by exp(ipx) where p is momentum, conservation of momentum is also built in because exp(ipx) is orthogonal to exp(ip1x). For <exp(ip1x) A W> where W is a wavelength only results consistent with conservation of momentum occur as is well-known in the literature. We argue that quantum mechanics of a free particle is statistical and linked to the idea of an equilibrium for which one does not consider time. Thus for exp(-iEt+ipx), t and x may be considered as independent. This implies that exp(ipx) must carry the idea of conservation of momentum by itself because one is not following a particle via x(t). “X’ may take on any value so this is an equilibrium type of scenario and the orthogonality of different exp(ipx) vectors ensures conservation of momentum. Thus a wave approach again implicitly contains the notion of conservation of momentum.
Newtonian mechanics (from the 1600s) defines a quantity called momentum p in terms of mv where m is mass and v=dx/dt with dx and dt tending to zero. Newton’s first law is equivalent to conservation of momentum in the absence of a force. The second demonstrates how p changes under a force. If one has equal action/reaction then overall momentum is conserved. As a consequence the idea of a “wave” associated with p does not arise. The main focus is on the idea of force. Fermat’s Least Time Principle for light (1600s), however, is based on minimizing the overall time traveled by light which reflects or refracts. Young’s interference experiments did not occur until the very early 1800s and Newton considered light to be a particle at this time. In fact light bounces off a foil creating an impact like a particle. Fermat’s principle does not directly make use of momentum or its conservation. It is focused on minimizing time. To do so, however, one must take a derivative with respect to a certain variable and set it equal to 0. This variable is the spatially invariant one in a problem. For example consider reflection (stationary or moving mirror) or refraction in two dimensions x,y. Let the mirror or medium face lie along the x axis at y=0. There exists a constraint on y and given a fixed starting point one cannot change y. This leaves x to be varied because it is invariant. Thus the extremization of time is really a statement of invariance of a spatial variable. In other words, Fermat’s principle is linked to the physical variables t and x while Newton’s focuses on p (momentum). This invariance in turn gives rise to conservation of momentum in the direction of the invariant variable i.e. (px). There is still the idea of Newton’s change in momentum as (py) abruptly changes in refraction as it hits the interface of media. The point we wish to make is that from Fermat’s least time principle one sees that conservation of momentum in the absence of a force is completely intertwined with invariance in space. The two cannot be separated in this approach and there is nothing unphysical about t or x. In fact, one may create a function A= -Et + (px)x + (py)y such that dA/dx partial = px = conserved quantity. In this function (px) and x are on the same footing. One does not have one without the other. This idea is carried over into special relativity as -Et+p dot r is a Lorentz invariant. Given that p momentum is physical (it manifests itself in impulse hits) and p and x are inseparable in (px)x we suggest there should exist a physical manifestation in x i.e. in space linked to the maintenance of conservation of momentum in this direction even in the absence of a force. This we argue leads to the physical idea of a wavelength which follows from A=-Et+px with A=0 in all constant velocity frames leading to a frequency proportional to 1/E and wavelength to 1/p. We have discussed these ideas in previous notes, but the main point we wish to make here is that equivalence of spatial invariance in the x direction in Fermat’s principle with conservation of momentum in the same direction suggests that if p is physical, the spatial invariance must also be manifested physically. It is not simply a mathematical convenience to link t to x to px, we argue. Furthermore this wavelength should be linked to establishing conservation of momentum as it does in quantum mechanics where the orthogonality of exp(ipx) i.e. Integral exp(-i p1 x) exp(ip x) dx automatically ensures p1=p without imposing momentum conservation a priori as in Newton’s first law. Both Newtonian mechanics and Fermat’s principle suggest that p may change abruptly due to a force or an immediate change in medium. The idea of a physical wavelength, however, suggests there may be interactions completely different from an abrupt change in p due to an impulse. In particular there may be interference or Bohm-Aharonov interactions. Newtonian Mechanics
In Part I we noted that Fermat’s least time principle was equivalent to conservation of momentum in the direction of an invariant spatial variable and that this principle is contained in the dispersion relation E=pc for a light which is strictly a wave relationship. We then argued that this wave relationship should somehow be extended to a particle with rest mass because such a particle may bounce off a mirror like light. To make the connection we wrote 0 = -Et+(px)x +(py)y for light with r/t=c and constant = -Et+(px)x + (py)y for a free particle with rest mass. In both cases Fermat’s principle is contained and is equivalent to conservation of momentum in the x direction. The form -Et+px, however, suggests a wavelength proportional to 1/p and a frequency proportional to 1/E for both light and a free particle with rest mass. We argue in this note that the wavelength represents spatial information which may show conservation (e.g. of momentum in a particular direction). Thus the focus of Part I was to link a free particle with rest mass to the wave formalism of light. Newtonian mechanics describes a free particle with rest mass as well as one moving in a potential without any need for a wavelength or frequency. In fact, for light E=pc means p→pn and c→c/n where n is the index of refraction means that E remains constant. For a particle with rest mass (nonrelativistic) E = pp/2m + V(x). A change in p means one needs the potential to absorb or yield energy to keep E constant which is different from light. Given that Newtonian mechanics describes particles with rest mass one may ask: Why bother to try to link a free particle with light and Fermat’s principle as done in Part I? The answer is that a free particle (with rest mass) diffracts or shows interference just like light in experiments. Furthemore -Et+px which suggests a wavelength proportional to 1/p and frequency proportional to 1/E is equivalent to the free particle action: -mo t sqrt(1-vv/cc) (relativistic) and .5 mo t vv (nonrelativistic) with v=x/t. Thus if Lagrangian/Hamiltonian formalism is identical to Newtonian mechanics we argue that the idea of wavelength and frequency is already present. It is not manifest because one deals with averages in pp/2m + V(x) = E. To see how wavelength may be incorporated in Newtonian conservation of energy, consider a free particle in a potential i.e. E= pp/2m + V(x). Next consider knocking out the particle so it has constant momentum. We argue that such a particle should exhibit wave-like behavour i.e be described in a manner linking it with light/Fermat’s principle e.g. be described by exp(ipx). Fermat’s principle already suggests that d/dx → p so -id/dx exp(ipx) = p exp(ipx) is an eigenfunction. We then argue that if exp(ipx) describes the particle when it is knocked out of the bound state, this information should still exist while it is within the bound state. Thus we suggest the form: W(x)= Sum over p a(p)exp(ipx) as describing the particle within a bound state. On average one has Newton’s conservation of energy result, but the idea that a particle may be knocked out and should behave as a wave is also present. Furthermore the various exp(ipx) interfere so the wave nature is also present in the bound state.
In Part III we considered the quantum free particle wavefunction exp(ipx) as consisting of two parts cos(px) and sin(px) treated as two different dimensions. We also argued that each, is a probability distribution in space and so associated with an internal force. One may notice that exp(ipx) does not contain time, but still predicts the direction of motion. Given that one cannot consider x(t) within a wavelength hbar/p even though on average the particle moves as x=p/m t, there is no real sense of time within a wavelength. Extending this idea, one repeats cos(px) humps. Given that the particle is moving, one must somehow juxtapose two spatial probability densities. The derivative of the distribution is proportional to average momentum and so one may see the notion of action-reaction occurring between cos(px) and sin(px) and within cos(px) and sin(px) accounting for the sign changes with x. We argue that d/dx cos(px) regions which change sign imply that the impulse direction changes. Furthermore, cos(px) has regions near the endpoints of the wavelength which have very low values (close to zero) which is fine if one has two juxtapositioned distributions. If, however, one creates a single particle bound state, exp(ipx) and exp(-ipx) for a(p)=a(-p) (a specific parity) give the appearance of a cancelation leaving the impression of 2cos(px) i.e a single “static” distribution. This occurs because there is no internal time in the picture and so p and -p probabilities add. The so-called canceled pieces have not really disappeared. as will be argued, they simply do not contribute to an average impulse which is what experiments apparently detect. Thus “negative” quantum probability is linked to impulse in the opposite direction, we argue. Thus interference is really linked to the notion of probabilistic impulses i.e. p exp(ipx) even though exp(ipx) is used as a kind of probability. exp(ipx) contains internal action-reaction impulse hits which have experimental consequences as we argue is shown in a reflection/refraction problem, but are also linked with impulse hits associated with p. Thus there are two sets of forces to deal with. This, we argue, explains why one writes the energy conservation equation AA/c1=BB/c1+CC/c2 for reflection/refraction where A,B,C represent the incident, reflected and refracted rays and AA simply shows that intensity/flux is positive as two equations: A+B=C and Ap - pB = p2C with E=pc. The two equations,, needed to solve for B,C in terms of A,, describe the internal force in exp(ipx) and the probabilistic impulse p exp(ipx). For a single particle bound state, one considers a linear combination of free particle states indicating that impulse hits represent the underlying physics. Given the apparent spatial invariance of a free particle exp(ipx), one may wonder how a kinetic energy profile appears. We argue that different p values yield different peak points for maximum impulse hits and these are out of synch leading to a spatial distribution of KE(x) or d/dx KE(x) which is equivalent to -dV/dx =Force where V(x) is the potential.
In classical statistical mechanics, e.g. a Maxwell-Boltzman (MB) gas, one calculates pressure by multiplying 2p (impulse) by v (velocity or flux). This is done because one has multiple particles and so flux is important. One needs to know how many particles strike a surface per sec as well as their impulse. What happens if one has a single photon (or particle)? This is not a steady stream situation, although one may create a corresponding steady stream situation. In particular, a single photon moving in the positive x direction may reach an interface of two media with different indices of refraction and either reflect or refract. In such a case one has a probability to reflect and one to refract which add to 1. Thus there are two unknowns (i.e. the two probabilities) and energy stays constant. One may immediately write a conservation of energy equation and apply it to a steady state scenario. This necessarily utilizes the idea of velocity because there is a flux. This energy conservation equation may be rearranged to give the appearance of a pressure balance equation which also depends on velocity i.e. flux. The problem is that conservation of energy or pressure balance is a single equation and there are two unknowns. One may not simply say that the probability to reflect is .5 and that to refract .5. In fact, in a previous note (1) we solved this problem using the idea of writing the energy equation in terms of two equations linked to differences of squares. We suggested that these two equations which only involve weights are linked to a function which together with its first derivative is continuous at x=0 .In (2) we suggested that the x-distributions cos(px) and sin(px) represent action-reaction impulse type distributions. We argued that an impulse creates momentum p in the first place and that this “impulse”, in the form of x-distributions, is still present because the object may strike a foil and impart impulse at any time. Furthermore, because cos(px) and sin(px) are probability distributions, there is no sense of time within a wavelength. On average, however, the object moves with a constant velocity. In the case of light it is c/n where n is the index of refraction and in the case of a nonrelativistic particle p/mo. The example of a single photon either reflecting or refracting allows one to formulate the problem in terms of impulse i.e in terms of the probability distribution scheme such that one does not have time and hence velocity. This scheme requires two equations. This impulse based x-probability distribution scheme or quantum mechanical approach, however, should map into the classical pressure balance because on average a photon does move as x=c/n t which it does. It is, however, the quantum approach (impulse picture with no time or velocity within a wavelength) which solves the problem i.e. yields the probability to reflect or refract which may then be linked to classical type (steady stream intensities and flux.
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.