In Newtonian mechanics, a space-dependent potential V(x) is associated with both a change in energy in x and a conservative force -dV/dx. There is no notion of an interaction through a resonance, even though classically resonance situations exist. A quantum free particle is represented by the space and time two-dimensional probability distributions exp(ipx) and exp(-iEt) with x and t representing independent variables, even though on average there exists the relationship x= p/m t. In the case of the time-independent Schrodinger equation, one uses a potential V(x) taken from “classical” physics which is combined with the nonclassical exp(ipx) (i.e. with time removed). An immediate issue, we argue, is that V(x) implies different potential energy values at each x, while exp(ipx) → cos(px) describes probabilities at different x points associated with the same p. According to classical mechanics, p should change from one x point to another in a deterministic manner governed by V(x). Thus there exists a contradiction. We argue that the probabilistic form cos(px) remains intact, but that the interaction between an exp(ipx) and V(x) involves the formation of a probability resonance which involves other exp(ipx)s in the problem. In other words, V(x)=Sum over k Vk exp(ikx) suggests that the potential also behaves in a probabilistic manner and one may have Vkexp(ikx) a(p-k) exp(i (p-k) x) = Vka(p-k) exp(ipx). One may argue that Sum over k Vk expIkx) is simply a mathematical formalism (namely a Fourier series) without any physical manifestation. In this note, however, we argue against this and suggest that the “resonance” involved is linked with physical probability distributions which exist in both V(x) and external energy-supplying objects (such as a photon). According to the time-independent Schrodinger equation there are discrete energy levels associated with this differential equation, so an entity which may provide energy is required to lift a quantum bound particle from one energy level to the next. Electrons in an atom absorb photons to move to higher levels. Thus the photon is an energy-supplier and the fact that it has a spatial wavelength does not seem to directly enter into the solution of the time-dependent Schrodinger equation. If, however, a particle may only interact through a probability resonance, we argue that it is not only the potential V(x) which must physically be able to interact in a probabilistic way, but also any energy entities used to supply energy. Thus we argue that a photon must have a spatial distribution i..e wavelength to allow it to be part of the formation of spatial resonances in the particle - V(x) system which it in fact does. Thus we argue that the wavelength of a photon is experimental evidence for x-probability distribution resonances which occur between V(x) and a bound particle.
In Newtonian mechanics, a space-dependent potential V(x) is associated with both a change in energy in x and a conservative force -dV/dx. There is no notion of an interaction through a resonance, even though classically resonance situations exist. A quantum free particle is represented by the space and time two-dimensional probability distributions exp(ipx) and exp(-iEt) with x and t representing independent variables, even though on average there exists the relationship x= p/m t. In the case of the time-independent Schrodinger equation, one uses a potential V(x) taken from “classical” physics which is combined with the nonclassical exp(ipx) (i.e. with time removed). An immediate issue, we argue, is that V(x) implies different potential energy values at each x, while exp(ipx) → cos(px) describes probabilities at different x points associated with the same p. According to classical mechanics, p should change from one x point to another in a deterministic manner governed by V(x). Thus there exists a contradiction. We argue that the probabilistic form cos(px) remains intact, but that the interaction between an exp(ipx) and V(x) involves the formation of a probability resonance which involves other exp(ipx)s in the problem. In other words, V(x)=Sum over k Vk exp(ikx) suggests that the potential also behaves in a probabilistic manner and one may have Vkexp(ikx) a(p-k) exp(i (p-k) x) = Vka(p-k) exp(ipx). One may argue that Sum over k Vk expIkx) is simply a mathematical formalism (namely a Fourier series) without any physical manifestation. In this note, however, we argue against this and suggest that the “resonance” involved is linked with physical probability distributions which exist in both V(x) and external energy-supplying objects (such as a photon). According to the time-independent Schrodinger equation there are discrete energy levels associated with this differential equation, so an entity which may provide energy is required to lift a quantum bound particle from one energy level to the next. Electrons in an atom absorb photons to move to higher levels. Thus the photon is an energy-supplier and the fact that it has a spatial wavelength does not seem to directly enter into the solution of the time-dependent Schrodinger equation. If, however, a particle may only interact through a probability resonance, we argue that it is not only the potential V(x) which must physically be able to interact in a probabilistic way, but also any energy entities used to supply energy. Thus we argue that a photon must have a spatial distribution i..e wavelength to allow it to be part of the formation of spatial resonances in the particle - V(x) system which it in fact does. Thus we argue that the wavelength of a photon is experimental evidence for x-probability distribution resonances which occur between V(x) and a bound particle.
A single particle bound wavefunction may be written as a Fourier series W(x)exp(-iEt) = exp(-iEt) Sum over p a(p) exp(ipx). We argued in previous notes (1) that exp(ipx) is a statistical free particle conditional probability whose modulus of 1 corresponds to P(x)=constant. For the case of a potential V(x), we argue it is stochastic and write it as: Sum over k Vk exp(ikx). The stochastic potential gives rise to a momentum distribution: P(p/x) = a(p) exp(ipx) / W(x) at each x. A resonance, however, seems to occur because writing W(x) and V(x) in the time-independent Schrodinger equation as Fourier series, yields: pp/2m a(p) + Sum over k Vk a(p-k) = E a(p) ((1)) Thus, a(p) is really a combination of other a(p-k) or, in other words, there is a correlation between different free particle momentum states. If such is the case for single particle interactions with a stochastic potential, we argue it is possible a correlation may exist for two particles. In other words, a stochastic potential may not necessarily produce two separate resonances for two quantum particles such as ((1)), but may instead form an overall resonance involving both momenta p1 and p2 “interacting or communicating” with Vk as in the case of entanglement. We attempt to investigate these ideas in this note.
In a previous note (1) we argued that a potential V(x), which exists in Newtonian mechanics, may be written as Sum over k Vk exp(ikx) (i.e. in a Fourier series). In another note (2) we suggested that the free particle wavefunction form exp(ipx) may be obtained from special relativity i.e. from the Lorentz invariant -Et+px. Thus in (1) we argued that the potential delivers impulse hits in a quantum mechanical scenario. In this note we revisit the emergence of exp(ipx) from special relativity, which like quantum mechanics, does not include acceleration i.e. a frame moves at a constant speed. (In Newtonian mechanics a particle moving within a constant dx is also said to have a constant speed, but this speed is differs in the next dx.) Given exp(ipx) in the context of special relativity, we consider the possibility of a system of impulse hits which may change p to p1 etc i.e. a set of exp(ipx)s. This leads to an average <pp> = {Sum over p a(p)exp(ipx) pp } / {Sum over p a(p) exp(ipx)}. A free particle satisfies: EE = pp + momo (c=1). If one wishes to retain a constant E (energy), one may replace pp with <pp> and imagine a function V(x) such that -(E-V(x)) (E-V(x)) + <pp> - momo. With p→-id/dx one obtains the Klein-Gordon equation. Taking a nonrelativistic limit yields the Schrodinger equation. We suggest the notion of potential may appear directly from special relativity together with the notion of a px dependent dynamic probability, namely exp(ipx) which is a quantum mechanical function. We further argue that the Newtonian approach ignores the information contained in -Et+px and uses a smoothing procedure.
We have argued in previous notes that both Newtonian mechanics and free particle quantum mechanics arise from special relativity. Thus, special relativity gives an expression for free particle energy and momentum and these are related to concepts of work. The Lorentz invariant -Et+px shows that x,t and x+hbar/p, t+hbar/E give the same solution to this potential-like term showing that there is stochasticity in the position and time delivery of interaction hits. In other words, even though the center-of-mass moves as x=vt, a momentum hit (proportional to rest mass mo) does not necessarily act at the center-of-mass. This gives the impression of shifts in probability, kinetic energy etc in the case of interference, i.e. interactions. For a single free particle, there is no interaction and -1/2m d/dx d/dx exp(ipx) = pp/2m exp(ipx). We argue that exp(ipx) is a probability and that in general one may add independent probabilities in an OR situation as in a 2-slit interference case. A bound state, however, is different because even though one adds weighted exp(ipx)s, they are part of a resonance or equilibrium and this is no longer an independent OR situation. We suggest that one uses a conditional probability when computing a Newtonian nonrelativistic conservation of energy equation to try to remove as much of the quantum features from this equation. As a result, one has: KEclassical(x) + V(x) = E = Sum over p pp/2m P(p/x) + V(x). One cannot bypass the fact that a quantum particle interacts differently from a classical one and that this involves a probabilistic granularity (wavelength), but classical notions of energy and energy conservation still exist. As a result, the bound state wavefunction Wn(x) = Sum over p an(p) exp(ipx) is not a usual OR sum, but is rather 1 over a normalization factor for an equilibrium picture. As a result, one cannot use Wn1(x)Wn2(x) = W(n1+n2) (x) as one would for independent probabilities. Furthermore, given that one has Wn(x) and Wn1(x)+Wn2(x) which are both of the form Sum over p a(p)exp(ipx), and both represent average energies, one needs a way to distinguish a resonance/equilibrium from a sum of resonance/equilibrium normalization factors. We suggest that the orthogonality of resonance Wn(x)’s provides this feature. The standard textbook explanation is based on Hermitian operators, but in the nonrelativistic case, we argue that one may obtain the result based on integration by parts based on the form of the p operator = -grad. Thus, it is ultimately the form exp(ipx) which leads to orthogonal bound state Wn(x)’s, together with the boundary condition that Wn(x)-->0 of x-> +infinite and -infinite.
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).
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.
In classical physics, there is a potential V(r) for a conservative force and force is linked to changes in momentum through Newton’s second law. For the sake of argument, imagine one may write V(r) as a Fourier series i.e. Sum over k Vk exp(ikx) and attribute physical relevance to this series. In other words, V(r) is a kind of average of microscopic granular system in which each exp(ikx) can impart momentum on a quantum particle. In addition, the Fourier series seems to have the appearance of a vector with exp(ikx) as a basis. It may also be noted that exp(ikx)exp(ipx) = exp(i (k+p)x). Thus, the potential seems to consist of ‘particles’ represented by exp(ikx) which may merge with quantum particles represented by exp(ipx) to form a boosted quantum particle exp(i (p+k)x). At first, it may seem exp(i (p+k)x) represents increased kinetic energy, but actually this merge should create potential energy. This perhaps gives a microscopic picture of potential energy. Thus, a merge of a Fourier series of V(x) with an ensemble of quantum particles Sum over p fp exp(ipx) yields: V(x) Sum over p fp exp(ipx) ((1)) A point which arises, however, is that Sum over p fp exp(ipx) cannot represent physical density because it may be negative at certain x . Thus, ((1)) is a departure from classical physics in which the potential energy is V(x) for one particle or V(x) density(x). ((1)) represents a kind of potential energy interaction, but introduces a new ensemble object Sum over p fp exp(ipx) which does not seem to exist in classical physics. This ensemble also seems to be a vector. The objective of this note is to try to determine properties and conditions which arise from this merge. In particular, we wish to find an equation satisfied by W(x) and also see how the classical picture is linked to this alternative scenario.