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 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.
Addendum Oct. 26, 2022 In ((3)) one should formally subtract i.e. d/dx{ -E1 t + (p1x) x +E2t + (p2x) x) = 0. Then p1x-p2x=0 and -E1 dt1/dx + E2 dt2/dx = 0. Thus conservation of momentum implies Fermat's principle. The minus sign between dt1/dx and dt2/dx follows from t1 being a variable of x and t2 of (L-x) where L is the length of the mirror such that initial and final y values have the same magnitude. (This does not change the conclusions made in the note we argue.) In a previous note (1) we argued that Fermat’s least time principle applied to light involved finding an expression for time in terms of spatial variables and the taking of d/dx where x is an invariant spatial variable in the problem. For example, for reflection from a stationary or constantly moving mirror in the y direction or refraction with different n(i) (indices of refraction in the y direction) x is invariant because there is no medium discontinuity in the x direction. We argued that Fermat’s principle is equivalent to a conservation law in the direction of the invariant variable and that this conservation is in fact that of momentum in this direction. In (2) we argued that E=pc with c=x/t i.e. the wave dispersion equation for light contains Fermat’s principle because one may write it as 0= -Et+px and apply d/dx using t(x) with x being the invariant variable. With conservation of momentum in the x direction and E1 = E2 for the two rays: dt1/dx + dt2/dx = 0 for a two ray problem. Fermat’s principle and E=pc pertain to light. The latter may be written as 0= -Et+px ((1)) for light with x/t=c which as we argued above leads to Fermat’s principle if one takes d/dx for two rays. One may note, however, that one also obtains Fermat’s principle for constant= -Et+px with momentum conservation in the x direction. Furthermore a particle with rest mass bouncing off a floor may behave like light. This we argue leads to the idea of considering two different frames. For light -E1 t1 + p1 x1 = 0 = -Et + px with x/t = x1/t1 = c. For a particle with rest mass -mo t1 + 0 = -Et+px with x/t=v where c=1 and mo is energy in a rest frame. (This is in fact the idea of a Lorentz invariant.) Thus Fermat’s principle for a free particle seems to appear in the form -Et+px = constant. The constant is 0 for light. The form -Et+px is consistent with a wavelength proportional to 1/p and a frequency to 1/E not only for light, but for the particle with rest mass. We argue that light moving in a medium is a complicated statistical process of scattering, but that the problem is treated in a “thermodynamical” manner by characterizing the medium by a single number i.e. single piece of information, namely the index of refraction i.e. c→c/n and p→pn for light. We argue that movement within the medium is like moving in a kind of constant potential for light. -Et+ (pn) x still holds and there is still conservation of momentum in the x direction for two media. Thus the idea of d/dx → p is already present in the photon form -Et+px which follows from Maxwell’s equations, but could also be derived from Fermat’s principle as argued in (2). In other words, ideas of quantum mechanics for a particle with rest mass could ultimately follow from Fermat’s principle, we argue. A free particle with rest mass is linked with constant= -Et+px and if one wishes to have an eigenfunction of momentum then: -id/dx exp(ipx) = p exp(ipx). For a constant potential, just like with light, p would change, but the form exp(ipx) would remain. Thus it seems that even in average situations (like refraction) one still wants to have a periodic type spatial function W(x) such that -d/dx d/dx W(x) /W(x) = <pp> such that <pp>/2m + V(x) = E for a particle with rest mass. This is the Schrodinger equation and it is found mathematically that only certain En are allowed if one wishes to have specific boundary conditions like W(x) →0 at +/- infinite for say an oscillator problem. Thus we argue that the fundamental idea of taking a derivative of an invariant spatial variable in Fermat’s principle which automatically yields conservation of momentum is a basic idea found in quantum mechanics even for a particle with rest mass, namely that d/dx is associated with momentum. Mathematically d/dx is a translation operator and so the link with Fermat’s principle shows how translation is linked with a spatially invariant spatial variable e.g. x. Furthermore Fermat’s principle shows that even complicated statistical scattering problems such as refraction (in which the photon scatters off of different molecules in different media) can be reduced to a periodic average problem with p→pn. This suggests that for a particle with a rest mass in a potential V(x) one should also have a periodic spatial variable such that d/dx d/dx is linked to average momentum, we argue.
In a previous note (1) we argued that Fermat’s principle of least time is present in the photon wave dispersion relationship E=pc which may be written as 0 = -Et+px with x/t=c. We argued that a nonzero constant = -Et+px applies to a free particle with rest mass, but that Fermat’s principle is still present. Furthemore the form -Et + px implies the existence of a frequency and wavelength for both a photon and free particle with rest mass. (Interestingly -Et+px is both the free particle relativistic action -mo t sqrt(1-vv/cc) and nonrelativistic .5mo t vv with c=1 and v=x/t. ) Fermat’s principle implies that d/dx → momentum (x component) as discussed in(1). An eigenfunction exp(ipx) demonstraties wave behaviour of a free particle with rest mass. In (1) we argued that if one knocks out a quantum bound particle it will be in a state exp(ipx). Thus this wave type information should also be included in the bound state so wepostulated a bound state represented by W(x) = Sum over p a(p)exp(ipx) (i.e. a wavefunction) such that the Fermat result d/dx → p still holds. In this note we wish to consider the idea of a statistical equilibrium and the concept of a constant energy. In the case of refraction it seems a complicated statistical scattering process occurs which slows down the motion of a photon i.e. c changes to c/n where n is the index of refraction. Thus one piece of information, namely n, describes the complicated statistical process as in thermodynamics. If E (energy) remains constant p→ pn. Thus in this complicated statistical scenario a cos(-Et+pn x) electric field pattern remains but p → pn i.e the wavelength has changed. The idea of a constant energy may even be applied to light scattering from a mirror moving with a constant v in the negative y direction. The actual energy and momentum of the incident and reflected rays are not equal, but for E1 special = p1 v(relative 1) and E2 special = p2 v(relative 2) with v(relative 1) = c- vcos(A) and v(relative 2)= c+vcos(B) where A and B are angles of incidence and reflection measured with respect to the y axis. E1 special t=E2 special. In this note we retain the idea that a particle knocked from a quantum single particle bound state is described by exp(ipx) in keeping with Femat’s principle. We consider the bound state to still have waves as Fermat’s principle suggests d/dx →p, but they cannot be of the same height and width because of the changing value of V(x) with x (i.e. the potential). According to Newton’s law one has conservation of energy i.e. pp/2m + V(x) = E, but we consider this to apply to a p (rms) = m v(rms) in the Schrodinger equation. If the forms exp(ipx)’s are to be retained within the bound quantum state (i.e. this information is not lost) and one has a constant E (energy) then the potential must be modified statistically to allow for the idea of conservation of energy for each p, but must also represent a constant potential to retain the form of exp(ipx). Thus in short the Schrodinger equation becomes a weighted sum of exp(i p(j) x) functions with each having a different wavelength (like refraction), but interacting with different constant potentials to ensure that E remains constant for each p(j). In other words we think of a potential V(x) as a statistical set of refraction discontinuities i.e. impulses which produce exp(ipx)’s which are like light moving through a homogeneous medium i.e. a momentum eigenstates. For this to happen, however, the various exp(ipx)’s need to combine with V(x) = Sum over k Vk exp(ikx) terms to create a Vmodified(p) exp(ipx) where Vmodiied(p) is a constant because each free particle exp(ipx) is associated with a different effective potential and a kinetic energy pp/2m, but the same energy En. We argue that this results from applying ideas of Femat’s principle to a particle with rest mass. Newton’s picture of a particle accelerating i.e. x(t) emerges if one considers rms values for which all information of exp(ipx) and crests/troughs of W(x) are ignored. In particular, W(x)=wavefunction may have crests and troughs, but -1/2m d/dx dW/dx/ W = KE(x) Newton which shows no crests or troughs, but simply balances V(x) in KE(x)+V(x) = E.
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
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 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 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.