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 (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.
Newton’s second law dp/dt = F(x)=-dV/dx may be written in terms of x alone i.e. dp/dx v(x) = -dV/dx or 1/(2m) d/dx pp = -dV/dx leading to the idea of kinetic energy pp/2m and potential energy V(x). In (1) we argued that there is velocity, rest mass mo, and momentum in quantum mechanics. A free particle, however, does not undergo smooth acceleration as in classical mechanics, and we argued there exists a momentum space wave exp(ipx) accompanying the moving particle. In such a scenario, force is considered to be probabilistic momentum hits of the form Vk exp(ikx) leading to a free particle momentum distribution W(x)= Sum over p a(p) exp(ipx). Both of these expressions involve interference due to momentum waves which do not track a particle in time. A mass flux mov or energy flux p=Ev relativistically seems to be the main consideration. Nevertheless, summing Vkexp(ikx) over k to obtain V(x) yields a function varying in x alone. This is a kind of average based on interference and Sum k Vk exp(ikx) is an average linked to momentum hits. We speculate that average momentum hits i.e. -d/dx V(x) are linked to momentum flux changes. In other words, even though an impulse is based on p and does not distinguish v i.e. p=m1v1=m2v2, momentum flux is velocity dependent and appears for an average momentum hit i.e. -dV/dx. In such a case (pave+dpave)(vave + dv ave) = 2 pave dv(ave) . This expression, however, contains both forward and backward motion which is part of quantum mechanics. Moving to an average picture seems to introduce time and direction i.e one uses pave d v(ave). This is already a change from a quantum to a classical scenario albeit for averages (which have not been specifically defined). Thus we consider pave dv(ave). If this is the case, then: d/dx pave 1/m space = -dV/dx or 1/(2m) pave pave = -d/dx V(x). Instead of treating pave pave as two separate averages one considers pave=prms or pavepave=<pp>. Thus the quantum bound picture (at any energy level) becomes a classical type of scenario which includes time and velocity. Note that for a two slit interference pattern, one does not need velocity and time because the interaction with the potentials from both slits is based on exp(ipx). One does not even need the form of the potential. Interference is the same for p=m1v1=m2v2 (even though the process takes place during different time intervals, the pattern is not affected). In other words, we argue that Newton’s second law might follow from a picture of momentum hits in which a flux i.e. p=mov is treated as a quantity which in an average scenario leads to a second flux i.e. an average momentum flux which reintroduces direction, time and velocity.. This is somewhat like a hydrodynamics scenario.
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.
The classical Lagrangian L leads to Newton’s second law which is equivalent to an energy-momentum conservation equation. The classical action: A = Integral dt1 L(t1) for a constant velocity v=X/T may be varied with respect X and T i.e dA/dT = -E and dA/dX= p. Thus, one may create a differential equation in (dA/dT)(dA/dt) and dA/dx dA/dx for the EE=pp+momo equation. Alternatively, one may create other differential equations. Quantum mechanical differential equations, however, are quite different with exp(i Action) being a solution. Even this may be written as the solution of a non-Schrodinger equation i.e. id/dt exp(iAction) = -1/2m (d/dx exp(iAction)) (d/dx exp(iAction)) for a free particle. For a constant velocity, exp(iAction) is equivalent to exp(ipx-iEt). We argue that this solution is statistical because it separates x and t (as independent variables) and holds for all x and t. Thus, quantum mechanics seems to be a statistical approach using free solutions exp(ipx-iEt) to create an ensemble as in classical statistical mechanics. A difference, however, is that exp(ipx-iEt) “interferes” even if one only takes the real part cos(px-Et). We consider the case of a particle in a box with infinite potential walls to argue that the periodicity is warranted i.e. positive and negative values (interference). Even though the modulus is 1, exp(ipx-iEt) may be operated on by various generators e.g. d/dx the generator of translations. This is analogous to operations on vectors in linear algebra. As noted above, quantum equations are not only energy-momentum conservation equations. They yield a wavefunction W(x) which is an unusual momentum distribution W(x)=Sum over p a(p) exp(ipx). exp(-iEt) holds overall because average E is the same at each x. Thus, one needs a W(x) which leads to a momentum-energy conservation equation, but also satisfies certain boundary conditions which become linked to spatial density= W*W. This leads to discrete energy levels and also cases of humps and troughs in W(x). We attempt to investigate some of these ideas in this note.
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.
Newton’s second law dp/dt = Force seems to be written in terms of a cause F(x) and effect p(x), because momentum is written in terms of a change i.e. d/dt. His third law, however,describes two causes i.e equal and opposite causes (forces) leading to two effects. We argue that a constant momentum, the effect of a force, may also act as a “force” if divided by a constant time i.e. change in p / dt(constant) = Force. A conservative external force may be written in terms of the derivative of a potential i.e. -dV/dx, but we suggest that a quantum free particle is linked with two distributions cos(px), sin(px) which describe an internal force/acceleration for even a constant p. The two combine in two dimensions i.e. exp(ipx) to create a type of probability which has a constant modulus i..e treats each x point equally. Each x point is treated equally because there is no external force -dV/dx, but we argue that there is still an internal one linked with probability and spatial distributions i.e cos(px),sin(px). If an impulse is applied to a quantum free particle to change it from p1 to p2, the change in energy is completely described in terms of momentum i..e p2p2/2m - p1p2/2m. There is no need to consider space, but at the same time there is a change in the two distributions cos(px) and sin(px) due to the change in p. Thus a constant p is linked with two spatial distributions which have physical consequences i.e. two slit interference etc. There is an internal distribution in space. If a quantum particle is placed in a potential V(x), there is now both an external potential, but also an internal one associated with each exp(ipx) if one assumes an ensemble of exp(ipx)’s is created i.e. W(x)=wavefunction = Sum over p a(p)exp(ipx). The point we make is that both the external and internal potentials are linked with x distributions, thus applying d/dx effects both, but Newton’s conservation of energy KE(x) + V(x) = E deals with changes in kinetic energy only linked to the external potential V(x). Formally, one may write: -1/2m d/dx dW/dx / W + V(x) + E, but we argue that external and internal potentials “interact” and we try to describe the situation considering V(x)W(x) to demonstrate the interplay of internal and external forces from the point of view of the internal force (probability) exp(ipx) and statistical mechanical loss of information i.e. equilibrium.
In a series of notes (1), we argued that the time-independent Schrodinger equation may be considered as a statistical conservation of average energy equation. We further argued that V(x)=Sum over k Vk exp(ikx) delivers stochastic potential hits which create a momentum distribution W(x)= Sum over p a(p) exp(ipx) at each x. The form exp(ipx) describes a free particle and a statistical distribution is made of such free particles. The form exp(ipx) has a modulus of 1 indicating that P(x)=constant. d/dx exp(ipx) = ip suggesting the change in conditional probability with space is related to classical momentum. Given a momentum distribution, each p at x occurs at a different time. Thus, one does not follow a classical particle in time x(t), nevertheless an average energy conservation equation applies. Taking the time-independent Schrodinger equation E W = -1/2m d/dx d/dx W + V(x) W and writing EW = i d/dt exp(-iEt), one obtains a time-dependent equation. We argue that if the time-independent case represents a classical conservation of average energy, the time-dependent equation should be linked to Newton’s equation of motion. This immediately leads to a problem because we argue one does not want x(t) for a statistical system. As a result, x in the time-dependent Schrodinger equation, which represented spatial position before, now represents a spatial variable that is independent of time. We consider the oscillator case and note that an amplitude is such a candidate where x(t)=XA(t). Thus, the time-dependent Schrodinger equation ultimately becomes an equation in A(t) with the amplitude cancelling, at least in the free particle and oscillator case. From such a consideration, one may work “backwards” and find a function for which the time-dependent Schrodinger equation (equating powers of X) becomes Newton’s second law. From this, one may show that exp(i Action) where Action = Integral dt1 (0,t) L where L is the classical Lagrangian is this solution with d/dX applying to X which is an amplitude, not x(t).