In (1) a particle with rest mass and momentum magnitude p1 moves from a region with a constant potential V1 into a region with a constant potential V2 with the interface being along the x axis. Overall energy is conserved and it is noted that conservation of the x-component is analogous to Snell’s law. ((1) goes on to use a variational approach to argue conservation.) In this note we begin with Newton’s idea that momentum changes in the direction of a force and that this force is -dV/dx where V(x) is a conservative potential. For constant V then, forces only arise from discontinuities in potential. We note that for a two dimensional problem (x,y) choosing a starting point of (x=0, y=Y) and changing the incident angle (measured from they axis, leads to different x values. Thus x is a piece of information in the two dimensional problem. In one dimension (along the x axis), however, momentum is conserved because there is no discontinuity in potential. In this one dimension there is x invariance and x should not be a piece of information. The question becomes: Is there a function in two dimensions that when applied to the two regions of this problem leads to x information disappearing? We consider an ending point (L-x, -Y) where L and Y are constant and note immediately that: x+ L-x=L i.e. a constant so that x disappears. Thus if x is multiplied by a constant (namely the x component of p) then the function A1=(px) x + (py)y meets the requirement i.e. A1+A2 loses all information about x because of the conservation of (px). As a result one sees the association of p and x which in Newton’s theory would normally not be combined. Furthermore if one uses the constraint that L=1/(px) (and drops the y=Y,-Y requirement), then (px)L = 1 which holds for all px values. This is linked to the idea of having one constant (namely 1 here) representing all constant motion as discussed in (2). In this way one may see a characteristic length proportional to 1/px emerge so that (px)(1/px) is the same for all constant motion. This characteristic length is the wavelength which appears in experiments. A second issue we wish to consider is the quadratic form of pp in both nonrelativistic E= pp/2m + V(x) and relativistic (E-V)(E-V) = pp + momo (c=1) energy conservation equations. We use the refraction type problem (two constant potentials) together with conservation of (px) = p sin(theta). We use Newton’s idea that force changes momentum in the direction of -dV/dx, (py) and hence p magnitude must change, but (px) must be a constant. Thus one may write: p1p1 - p2p2 = (p1yp1y-p2yp2y) = C1 ((A)). We have used p1x=p2x, but ((A)) leads again to this result so the general result p1p1=C2 should hold. A similar argument holds relativistically. Thus without integrating one is led to the idea of a quadratic p (i.e. p magnitude p magnitude) as part of a constant equation (becuase the potential is constant). This we argue is conservation of energy.
In (1) a particle with rest mass and momentum magnitude p1 moves from a region with a constant potential V1 into a region with a constant potential V2 with the interface being along the x axis. Overall energy is conserved and it is noted that conservation of the x-component is analogous to Snell’s law. ((1) goes on to use a variational approach to argue conservation.) In this note we begin with Newton’s idea that momentum changes in the direction of a force and that this force is -dV/dx where V(x) is a conservative potential. For constant V then, forces only arise from discontinuities in potential. We note that for a two dimensional problem (x,y) choosing a starting point of (x=0, y=Y) and changing the incident angle (measured from they axis, leads to different x values. Thus x is a piece of information in the two dimensional problem. In one dimension (along the x axis), however, momentum is conserved because there is no discontinuity in potential. In this one dimension there is x invariance and x should not be a piece of information. The question becomes: Is there a function in two dimensions that when applied to the two regions of this problem leads to x information disappearing? We consider an ending point (L-x, -Y) where L and Y are constant and note immediately that: x+ L-x=L i.e. a constant so that x disappears. Thus if x is multiplied by a constant (namely the x component of p) then the function A1=(px) x + (py)y meets the requirement i.e. A1+A2 loses all information about x because of the conservation of (px). As a result one sees the association of p and x which in Newton’s theory would normally not be combined. Furthermore if one uses the constraint that L=1/(px) (and drops the y=Y,-Y requirement), then (px)L = 1 which holds for all px values. This is linked to the idea of having one constant (namely 1 here) representing all constant motion as discussed in (2). In this way one may see a characteristic length proportional to 1/px emerge so that (px)(1/px) is the same for all constant motion. This characteristic length is the wavelength which appears in experiments. A second issue we wish to consider is the quadratic form of pp in both nonrelativistic E= pp/2m + V(x) and relativistic (E-V)(E-V) = pp + momo (c=1) energy conservation equations. We use the refraction type problem (two constant potentials) together with conservation of (px) = p sin(theta). We use Newton’s idea that force changes momentum in the direction of -dV/dx, (py) and hence p magnitude must change, but (px) must be a constant. Thus one may write: p1p1 - p2p2 = (p1yp1y-p2yp2y) = C1 ((A)). We have used p1x=p2x, but ((A)) leads again to this result so the general result p1p1=C2 should hold. A similar argument holds relativistically. Thus without integrating one is led to the idea of a quadratic p (i.e. p magnitude p magnitude) as part of a constant equation (becuase the potential is constant). This we argue is conservation of energy.
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 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.
Newton’s second dp/dt = Force where p is momentum is equivalent to pp/2m + V(x) =E for a conservative force= -dV/dx with p=mo v(x), v being velocity. In general one may divide the x axis into equally spaced dx units with dt(x) (i.e. time spent) differing in each unit. A main point we wish to make is that dt(x) is the same whether the particle moves to the right or to left. Thus using dt(x) as a kind of density (proportional to C/v(x)) as done in (1) means this spatial density does not distinguish between motion to the right or left. This is one kind of spatial density we argue, one that considers only the magnitude of velocity. Heuristically, however, one might argue that motion to the right or left involves a skewed spatial density in the direction of motion. This we argue represents a second kind of spatial density which follows from Newton’s first law and is associated with quantum mechanical motion. Ideally the two should be linked. Newton’s first law indicates that a particle at rest remains at rest and one in constant motion in a direction continues in this state unless acted on by force. This law is usually not associated with a mathematical equation and so a spatial density is not derived. In (2), however, we argue that Newton’s first law is associated with invariance in a spatial direction if there is no force in that direction, for example x. In such a case, d/dx is the translation generator in that direction and momentum is kind of “information”. We argued in (2) that one may write d/dx f(x) = p to describe this with the simplest solution being f(x)= px for constant p. We further argued that this is in fact linked to Fermat’s least time principle for light. Time =distance/speed of light, but momentum and displacement are parallel. Thus for a two dimensional problem: p dot r = (px)x + (py)y = p distance = p sqrt(xx+YY) and p sqrt( (L-x)(L-x) + YY) for two rays, one being the incident and the other the reflected. Given that there is conservation of momentum in the x direction one may write (px) incident = (px) reflected or d/dx (p distance incident) - d/dx (p distance reflected) = 0 which is Fermat’s principle for reflection with one index of refraction. For refraction p1=pn1 and p2=pn2, but (p1x) = (p2x). f(x)=px satisfies d/dx f(x) = p. This expression depends on x, but also on the direction of motion i.e. one may have p or -p. In (2) we further argued that this expression may be written in terms of: i px = ln(exp(ipx)) with exp(ipx) being periodic, but neither growing or tending to a constant zero for a constant p. As a result we identified exp(ipx) with a second type of spatial density or probability, one which accounts for the direction of motion. exp(ipx) is also associated with a quantum mechanical description of a free particle. Thus Newton’s first two laws give rise to two different spatial densities, but they are related. In particular, the modulus of exp(ipx) equals constant which is P(x) or spatial density from Newton’s second law. In this note, we start with the assumption that a velocity (not velocity magnitude) spatial density should exist and examine what properties of such a density/probability arise.
In (1) we argued that one may derive Newton’s second law (which is equivalent to an energy conservation equation) by using information arguments. We argued that V(x), a potential, is a fixed piece of spatial information linked to the x axis being divided into equal dx units. The variable spatial information is dt(x) i.e. the time spent in each dx or v(x) =dx/dt (to put this in the same form as the time displacement generator). One may not simply compare v(x) directly with V(x) because they change differently. v(x) changes in a weighted way i.e. v(x)d/dx or d/dt while V(x) changes as d/dx V(x). Thus we suggest d/dt v(x) = C V(x) which ultimately leads to Newton’s second law. Using d/dt → v(x)d/dx and integrating yields conservation of energy: p(x)p(x)/2m + V(x) = E. A similar argument was used in (2) to deal with the special relativistic case. There one cannot use d/dt → v(x)d/dx to obtain a conservation of energy equation. The point we wish to make is that for both the nonrelativistic and relativistic classical equations there are two types of “information” equations. The first is a “force” i.e. -dV/dx equation which is based on how V(x) changes with x. The LHS of this equation is linked to d/dt p(x). The second is a corresponding energy conservation equation. The quantum case is quite different, we argue, because a free particle is described by exp(-iEt+ipx). There is still the notion of p=mo dx/dt(x) as in the nonrelativistic case because p must be defined, but exp(ipx) is additional imposed piece of spatial information. -id/dx exp(ipx) = p exp(ipx) and -d/dx d/dx exp(ipx) = pp exp(ipx). This begs the question: How does one reconcile the two? For the nonrelativistic Newtonian and relativistic cases it is possible to have two equations (one with Force the other with energy) using one x dependent variable p(x). In the quantum case one may take an average to define a kind of complex = Sum over p a(p) p exp(ipx)/ Sum over p a(p)exp(ipx, but this is complex, not real. Furthermore both +/- is included in so there seems to be a problem linking it with changes in V(x) in one direction. Thus it seems that one is forced to consider the second kind of information equation from classical mechanics and define a <pp> = Sum over p a(p) pp exp(ipx)/ {Sum over p a(p)exp(ipx)}. In the nonrelativsitic case this may be multiplied by 1/2mo and then added to V(x) to obtain a constant E. If one only looks at <pp>/2mo as a spatial function then this is identical to KE(x) i.e. kinetic energy(x) for a classical system. Thus in the nonrelativstic case, Newtonian energy conservation is always embedded in the quantum system regardless of the energy. In other words the quantum bound equation i.e. the time independent Schrodinger is not obtained independently from the classical one in terms of information.One may ask: How does information related to exp(ipx) affect the quantum system? Two immediate ways are as follows. First <pp>/2mo = -1/2mo d/dx dW/dx / W where W= Sum over p a(p)exp(ipx). This leads to a differential equation which has discrete En values verified by experiment. Secondly W is linked to spatial density W*(x)W(x) which exhibits crests and near zero regions. Using <pp> suggests that p(x) in the classical case is linked to prms(x) quantum. Thus building quantum mechanics using “informational” ideas seems to involve embedding classical conservation of energy into the quantum equation a priori. This holds both nonrelativstically or relativistically (Klein-Gordon equation). This immediately raises the question: If classical mechanics (in the form of energy conservation) is embedded in quantum mechanics for any energy, why do people seek to take some kind of limit (3) to obtain classical mechanics from quantum? We argue that quantum mechanics has more spatial information than classical, but retains classical conservation of energy so there is no limit to take. One may simply ignore ideas related to the wavelength to obtain classical mechanics, we argue.
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.
Newtonian mechanics often employs a potential V(x) such that .5m v(x)v(x) + V(x) =E ((1)) where v(x) is velocity. Mathematically V(x) is defined at each x, but in reality there should be a tiny cutoff i.e. finite dx. Newtonian mechanics makes use of the idea of small dx lengths which tend to 0 mathematically. To first order v(x) is constant within a dx range, but the overall mathematics is continuous instead of discrete. Thus there is a continuous equation ((1)) for conservation of energy. A photon may be described by an eclectic field varying as cos(-Et+px). In terms of wavelengths exp(ipx) is often used as a mathematical convenience and noncontinous behaviour is not an issue. For example in refraction, momentum = hbar/wavelength abruptly changes and one straight line path is joined to another with a different slope. exp(ipx) is associated with repeatable wavelength patterns with a single wavelength representing a spatial distribution/ (kind of probability) region. A particle with rest mass may also be described by exp(ipx) which suggests a characteristic length hbar/p. Within this length (i.e at each x point which exist within hbar/p) the particle has p and pp/2m. Thus for a square well potential there is conservation of energy at each x point even though there is a wavelength in the problem. This is fine for a constant momentum, but what about a potential V(x) which implies p must change at each dx region? exp(ipx) is an eigenfunction of -id/dx which is linked to spatial invariance. It manifests itself physically by interference (e.g. two slits) which again involves non-smooth motion (i.e. one compares rays with completely different slopes). The characteristic length hbar/p is fixed and associated with a single p, yet momentum is supposed to change from one x point to the next. hbar/p does not fit within a single dx and this is a problem. Quantum mechanics remedies this issue by allowing for a superposition of exp(ipx)s i.e. W(x) = wavefunction = Sum over p a(p) exp(ipx) such that energy conservation is forced at each x. Thus one combines the idea of wavelength (which is physical) with energy conservation which is also physical. A consequence of this, however, is that -id/W/dx / W leads to a strange average momentum which is not the square root of -1/2m d/dx dW/dx/ W as well as discrete energies. A mathematical benefit is that one has a continuous differential equation, namely the time-independent Schrodinger equation. The question we ask is: What happens in the so-called classical limit? In the nonrelativstic case this leads to Newtonian mechanics in which there is no wavelength. How does the wavelength disappear so to speak? We suggest (as has been suggested often in the literature and textbooks) that the wavelengths in W(x) may become extremely tiny so as to not be detected easily in classical length scales. Mathematically the time-independent Schrodinger equation holds and there is a smooth W(x) at all points, but we suggest that this smoothness may not be necessary because it is not found in refracting light. If the wavelength is tiny compared to small dx regions in the classical limit, then many wavelengths of a given p fit into dx. Thus the idea of spatial invariance approximately holds within dx and there is no issue of the wavelength spilling out of the dx region as it does when it is comparable to the size of the system as in quantum bound problems. Thus within dx a single p with many (or at least several) wavelengths holds and there is a single p. The kinetic energy is pp/2m and the potential V(x) where x is in the center of dx. One may think of p changing abruptly (nonsmoothly) at the boundary of one dx with another such that conservation of energy holds. Thus one has p wavelengths which do not spill out of dx and give an approximate sense of spatial invariance together with energy conservation. The only loss is that of a smooth wavefunction, but it is not clear that such smoothness is a requirement. V(x) is physically associated with a limiting dx length (at which the concept of V(x) does not hold, but rather there are stochastic hits) and if one expands this length i.e. chooses a much bigger dx in keeping with classical scales, it seems that there should not be a problem. It as if V(x) delivers an impulse hit at each dx junction (like a series of different media in refraction).. This approach leads to the decoherence of wavelengths in the classical limit.