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.
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.
In previous notes (1), we argued that quantum mechanics follows from treating the potential in terms of certain physical units of the form exp(ikx) i.e. V(x)=Sum over k Vk exp(ikx). These deliver an impulse of momentum k. exp(ikx) has a number of properties. First, it is invariant in space as much as can be under the circumstances (periodic), secondly it is two dimensional with a modulus of 1 and thirdly, it has positive and negative portions. Furthermore, we argue that a quantum particle in a potential takes the same form i.e a distribution of exp(ipx)’s i.e. W(x)=wavefunction=Sum over p a(p)exp(ipx). In other words, it also acts like a kind of potential, but it is associated with an average kinetic energy. In this note, we consider the form exp(ipx) in terms of action/reaction. The quantum particle or unit of potential delivers an impulse hit and so should be linked to the idea of action reaction which is already known in classical mechanics. There, action/reaction is considered in terms of two objects. In this note, we speculate that the quantum particle or unit of potential may already have action/reaction built-in. This would describe why exp(ipx) is periodic (i.e has positive and negative portions while still having an overall modulus of 1). It would also help explain interference. Finally, one may note, as is well known in quantum mechanics, that the form exp(ipx) leads to conservation of momentum through <exp(ip1 x)exp(ip2 x) constant exp(ip3 x)exp(ip4 x)> i.e. a built in action/reaction exp(ipx) form is directly linked to momentum conservation.
The Schrodinger equation for a bound state includes the classical potential V(r) for which –d/dxj V represents Fj or force in the j direction. It has been argued in a previous note (1), that one may attribute physical meaning to the Fourier series of V(r). In such a case, V(r) and W(r) (the wavefunction) are quantities averaged over momentum (or vectors) as well as being averages over time. One may write the product of V(r)W(r) as one of two Fourier series and then reorganize into a series with basis exp(ikr). In such a case, one examines an interaction with V at each momentum wave level. Then the average is only over time as different k momentum states of in W combine with different p momentum states of the Fourier series of V(r). For a p momentum wave, one may write: p2/2m fp + Sum k Vk f(p-k) = E fp where W= Sum over p fp exp(ipx) in one dimension ((1)) The objective of this note is to observe differences in the interaction of a single plane wave with a potential ((1)) and the interaction of W with V in the Schrodinger equation. In particular, for a particle in a box with infinite walls, V(x)=0 inside the one dimensional box, yet the Fourier series of W includes all p values. From ((1)), even though V(x)=0 as an average, there seems to still be an interaction with various Vk values (Vk= the kth Fourier transform) in the interior of the box as ((1)) holds at all x points.
Mathematically, it is possible to write a potential V(x) in classical mechanics as a Fourier series. It is not guaranteed, however, that this series have any physical meaning. In this note, we argue that it has meaning if one thinks of the potential as an ensemble of different momentum sources, each with a different weight. Then, placing a particle in such a potential could yield an ensemble of momenta f(p). This ensemble W(x) at this point is not associated with a Fourier series, but it is an equilibrium ensemble and can also be thought of as a potential, it seems, which has different weights associated with different momenta. If this is the case, this ensemble can also be written as a Fourier series W(x) = sum on p f(p) exp(ipx). Now, the ensemble W(x) created by the potential V(x) in the first place, can interact with the potential again and again. This can be described by writing V(x)W(x). This combination should return the original ensemble W(x) up to a constant if one has an equilibrium situation. Now V(x)=Energy-average Kinetic energy which leads to the Schrodinger equation.
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 previous notes, we argued one may formulate bound state quantum mechanics by assuming a stochastic potential which delivers “hits” of momentum k V(k) exp(ikx), while having an average form of V(x). A main constraint seems to be that the different k V(k) exp(ikx) hits may occur at any x point and that V(x) be of the form Sum over k b(kx) so that -d/dx b(kx) = -k b(kx). In other words, conservation of energy exists at the level of each hit as well as for the overall average. To force all of these constraints, a solution of b(kx)=V(k) exp(ikx) emerges which introduces periodicity and complex numbers. It should be noted that V(x)= Sum V(k) exp(ikx) so each term is actually a “piece of potential energy”. Furthermore, we argue that V(x) is decomposed for physical reasons i.e. V(x) is a low resolution result which incorporates all of the “pieces”. In a quantum mechanical situation, a “piece” V(k)exp(ikx) may act at t1 and V(k2)exp(i k2x) at t2. We argue that exp(ikx) should have physical meaning and leads to sin(kx) terms for the “k” and “-k” pieces. Thus, there is forward and backward motion which “cancel” i.e. have nothing to do with V(x). For a particle, however, in this region, there may be forward and backward motion, both of which have kinetic energy which should not be part of the average kinetic energy at x. The various physical (resolved) pieces of the potential lead to momentum distribution at each x point with P(p/x)= a(p) exp(ipx)/ W(x) (W(x)=wavefunction). In this case, one has a probability, not “pieces” of KE(x). At first, one may be alarmed by complex probabilities, but they are necessary to allow for cancellation of kinetic energy pieces due to the positive/negative values combined V(k)exp(ikx) pieces. Thus, even though KE(x) (average) matches classical results, the actual motion is more complicated. The potential seems to be the driver of the ensemble picture and should be linked to the microscopic physical nature of the potential. In this note, we wish to examine features and consequences of such a stochastic potential, including the idea of tunneling and infinite potential walls.
The classical limit of quantum mechanics is often considered in terms of the time dependent Schrodinger equation. In this note we consider the bound case solution i.e. a time dependence of exp(-iEt) and as in previous notes argue that time has been removed from this statistical equilibrium picture as a the quantum particle tries to establish a kind of impulse potential field based on superpositions of exp(ipx) terms. Such a potential field is linked with a bound state and does not involve following the particle about in time as it undergoes stochastic impulse hits with exp(ikx) terms from V(x)=Sum over k Vk exp(ikx). On average a classical energy balance equation KE(x)+V(x) = En holds for every energy En, but the average is created by a spread of momenta i.e. W(x)=wavefunction= Sum over p a(p)exp(ipx). In this note we consider the case of a particle in an infinite potential well and note how the spatial resolution becomes very high at the same time that the momentum spread becomes very narrow in the case of high n (energy level). Thus as n becomes approaches infinite, it is not just the vrms(x) which is linked with x, but the statistical scenario of quantum mechanics approaches a situation in which x is being resolved to a very high level and p is almost constant. Furthermore quantum entropy becomes a constant (independent of n) as n→ infinite suggesting a theory which is nonstatistical in this limit.This would represent a classical picture except for the fact that time is still removed from the quantum bound state i.e. forward and backward exp(ipx), exp(-ipx) still interfere. Given that x and p are well resolved locally for high n it seems one may follow the high n quantum particle in space and time and manually remove the forward and backward portions to create a theory of a particle moving in one direction under V(x) i.e. classical physics. This bypasses the idea of a wavepacket whose x-spread increases in time and allows one to base classical physics on a bound state solution of a quantum particle in an infinite well.