In part I we argued that classical mechanics divides a one dimensional line into equally spaced dx regions. These are small, but not zero because classical spatial density is C1 dt = C1 dx/ v(x) where v(x) is velocity which is assumed to be constant within dx to first order. In terms of probability, if a particle is within a certain dx, it is not in the adjacent ones i.e a probability distribution reaches zero at the endpoints of dx. We argued that for a constant velocity classical density is P(x)=constant and so one must keep two spatial probability distributions (in two dimensions or separated by i) such that the modulus is 1. This makes the two ideas compatible. This idea seems to apply to quantum mechanics with the stipulation that dx is linked to 1/p. Thus one has exp(ipx) as a dynamical probability as discussed in Part I. In this note we wish to consider this in more detail together with the Schrodinger equation. In particular we focus on the high energy limit. We argue that exp(ipx) should hold for a small dx region classically which may include several wavelengths proportional to 1/p. Thus there is still the idea of the distribution and its shift as two dimensions, but we argue that the time-independent Schrodinger equation may be an approximation as it considers exp(ipx) and exp(-ipx) when creating Wn(x) and it is not clear that such interference holds in a very high n limit.
In part I we argued that classical mechanics divides a one dimensional line into equally spaced dx regions. These are small, but not zero because classical spatial density is C1 dt = C1 dx/ v(x) where v(x) is velocity which is assumed to be constant within dx to first order. In terms of probability, if a particle is within a certain dx, it is not in the adjacent ones i.e a probability distribution reaches zero at the endpoints of dx. We argued that for a constant velocity classical density is P(x)=constant and so one must keep two spatial probability distributions (in two dimensions or separated by i) such that the modulus is 1. This makes the two ideas compatible. This idea seems to apply to quantum mechanics with the stipulation that dx is linked to 1/p. Thus one has exp(ipx) as a dynamical probability as discussed in Part I. In this note we wish to consider this in more detail together with the Schrodinger equation. In particular we focus on the high energy limit. We argue that exp(ipx) should hold for a small dx region classically which may include several wavelengths proportional to 1/p. Thus there is still the idea of the distribution and its shift as two dimensions, but we argue that the time-independent Schrodinger equation may be an approximation as it considers exp(ipx) and exp(-ipx) when creating Wn(x) and it is not clear that such interference holds in a very high n limit.
A classical particle is considered to be a point described by x(t) i.e. x and t are both a set of points. In a one dimensional problem, however, dx= v(x) dt so if all dx tend to zero, then dt’s are small, but don’t tend to zero in the same way. In other words dx may be considered small, but of a finite size. To have a particle within dx, it should have a probability distribution that goes to zero at the endpoints of dx i.e. one could have a repeated set of these for a constant velocity. This, however, contradicts P(x)= C1 dt = C1 dx/v(x). Thus it seems first that there are two kinds of probability distributions and that the set of distributions fitting into each dx is incomplete because if this set is described by a function F(x), this function should have the same magnitude at F(x+dx) for a free particle. This suggests a phase function linear in x i.e. exp(i C1 x). P(x) = C1dx/v(x) is based on dynamics and no dynamics appears in exp(i C1 x). It seems that dynamics should occur leading to the possibility of exp(i p x). (This , however, is ad hoc.) We call this a dynamic probability as opposed to P(x). In quantum mechanics it is called a wavefunction. If one considers the classical free action (relativistic or nonrelativistic) with x and t independent, then A= -Et + px (1) and one may create eigenvalue equations -id/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = E exp(iA) so exp(ipx) also appears more formally. In other words a fixed resolution periodic distribution requires the same shifted distribution (i.e. cos(px) and sin(px)) kept as separate dimensions i.e. “i” in order to create a dynamic probability which is a phase function which may be linked to P(x). This phase function exp(ipx), however, consists of two classical-like waves showing that in each dimension probability is “carved” out of troughs and added to crests. The two dimensions allow for the dynamic probability to be consistent with P(x)=constant for a free particle.
In a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by exp(ipx).
In the literature, stochastic approaches employed to derive the Schrodinger equation seem to focus on the idea of a classical spatial density which depends on time and space ie. density(x,t). As a result, d/dt (partial) density appears in various equations such as the continuity and Focker Planck equations. At one point in the derivation the quantum wavefunction is introduced “mathematically” as W(x,t)=sqrt(density(x,t)) exp(i S(x,t)) where S is related to a velocity field through u(x,t)=velocity= d/dx S. From quantum mechanics, -Et becomes a term of S the velocity field. The stochastic/hydrodynamical approaches employ the term du/dt partial = d/dt (partial) d/dx S and then “pull out”” d/dx leaving d/dt (partial) S which yields -E average energy from the -Et term. In this note we argue that for a bound quantum state density(x) so d/dt (partial) density = 0, yet the statistical ideas of quantum mechanics should be present in such a case. Furthermore for the free particle wavefunction exp(ipx-iEt), density=1. In fact, the Schrodinger equation should follow for a free particle even the case of density=1. Stochastic processes seem to be based on equations which consider a change in classical spatial density at a point x in time due to flow in or out or stochastic changes. Quantum processes seem to deal with periodic probability changes from one dimension to another exp(ipx) (2 dimensions) which are somehow related to a physical process of momentum flow. Ultimately one links this process to an energy conservation which does not seem to be the same as a classical stochastic/hydrodynamical approach. In this note we argue that a free particle with a constant momentum p has periodic stochastic processes associated with it through its complex probability change in the region of its wavelength, but that these are linked to creating the momentum p. Furthermore momentum is linked to energy so we argue that the Schrodinger equation is a stochastic equation linking energy and momentum and that time and space are separated i.e. there is no situation of following a change in spatial density in time for a free particle with momentum p. For a bound state, one has a distribution of free particles, but again time is removed from the problem in the sense that it only appears in exp(-iEt) in the wavefunction to again establish a relationship between energy and average momentum and potential. There is again no nonzero d/dt (partial) density term which seems to be key in stochastic equations. Only if one takes a linear combination of energy eigenstates Sum over En an(En) exp(-iEnt) Wn(x) does one obtain the form density(x,t), but we argue that quantum features should already appear in exp(ipx) and the bound state scenario with a zero d/dt (partial) density(x,t). We argue, however, that the Schrodinger equation and fundamental stochastic ideas of quantum mechanics should already be present at the level of exp(ipx).
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.
The uncertainty principle (linked to p,x or E,t) plays an important role in quantum mechanics. For example, an electron with an exact momentum p is described by a wavefunction exp(ipx). This function exists at all x points and so it is suggested in the literature that there is complete uncertainty as to the electron’s position. (More specifically P(x)=exp(-ipx)exp(ipx) normalization constant=constant and so all x points have the same probability.) Thus if p is completely certain, it is said that position x is completely unknown. We note, however, that exp(ipx) consists of cos(px) and sin(px) which are repeated probability distributions (with zero endpoints). Thus even though they exist in all of space, due to repetition which in turn is due to spatial invariance, there is a specific length number, namely the wavelength. Thus we argue that in classical physics there is a set of points (x,p), but in quantum mechanics ( p , x with its dx interval), with the latter being equal to hbar/p. Then the only uncertainty is within the dx interval i.e. the wavelength and spatial density is linked to momentum i.e. there is dynamical probability. In particular, there is a frequency hbar/E and a wavelength hbar/p which define an internal clock and ruler and the two are independent. Thus within a wavelength one has a probabilistic picture (in order to have endpoints defining the dx interval) and no sense of time, yet on average time exists because of hbar/E and the particle moves in space with a constant v. This begs the question: How do time and space x arise in such a scenario? We speculate the following. Assume there exists a property called energy which defines an object. This object, however, may break into two different objects. As a result, one introduces a parameter called time to distinguish between the two systems (E) and (E1,E2). For a certain time range one has a single object, while for a later time interval, one has two objects. Each of the two objects in the second system, however, needs to be distinguished as well and so another parameter, namely space x is used for this. If the two objects existed at the same x and t, how would one know there were two unless another parameter were introduced? We use t and x as the distinguishing parameters and furthermore give these a measure i.e. dt and dx. We then argue that from these ideas one may show the notion of speed and direction (because there are two objects) as well as conservation of momentum arises. Given the notion of an interval, we argue it must be described by a probability distribution and we link this to constant/p and exp(ipx). We finally note that since spatial density is linked to dynamical probability, one must consider motion associated with a density which may lead even to collective motion.
In quantum mechanics, a free particle is represented by exp(ipx). It has often been noted that exp(-ipx) exp(ipx) =1 yields equal probability for any point x. In classical mechanics, one might argue that a constant real number suffices. The form exp(ipx) is also identical to the classical electromagnetic expression associated with light (i.e. a photon). In this note, we wish to begin with the idea of an ensemble of momenta at x using a real probability f(p,x) and create a conservation of energy equation i.e. KEave(x) + V(x) = E. This equation may be differentiated to yield a force equation with d f(p,x)/dx. If one argues that should appear in such an equation, one has a link between d f(p,x)/dx and . One finds that f(p,x), which we assume to be the same for p and -p so that pf(p,x)-pf(p,x)=0 as in classical statistical mechanics, may be generalized to f(p,x) + i g(p,x) with g(p,x) linked to d f(p,x)/dx and . In other words, quantum mechanics includes two components, one associated with V(x) and <pp/2m at x> and another with F(x) and . The Bohm Aharonov effect seems to justify such an approach because one may have V(x) non-zero and F(x) zero and still have physical effects, thus a potential V(x) and F(x) may both have separate physical consequences. Using cos(px) and isin(px) seems to track these two separate consequences through the separate consequences of and <pp/2m at x>. For a classical particle, p becomes pp by squaring so the two do not need to be treated with two separate “dimensions’ i.e. cos(px) and isin(px). In this note, we try to obtain the form exp(ipx) from the more general exp( i Theta(x,p)) without resorting to the form exp(ipx) of a photon in classical electromagnetism and also consider an interpretation.