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.
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 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.
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.
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 in quantum bound states, the classical potential V(x) is really an average i.e. equal to Sum over k V(k) exp(ikx). Mathematically, this is the well-known Fourier series, but in this note we try to argue there are important physical implications which lead to the hump like densities in numerous cases in quantum mechanics, unlike what is found in the classical case. In addition, we argue that the stochastic potential creates a momentum distribution at x which may be used to calculate average kinetic energy (at x). Because average kinetic energy changes with x, the momentum distribution must also change. At a given x, summing P(p/x) accounts for all possible momentum states and yields W(x), the wavefunction. Because this differs with x, the probability P(x) must differ with x as well, yielding a quantum density not seen in classical physics. Thus, quantum bound state density is due to the spatial change in momentum distribution probabilities with x, we argue.
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.
The momentum wavefunction fp of a particle in a box (with infinite potential walls) has been described (1) (2) in terms of many plane waves (i.e. many fp’s) even though the energy E=(hbar k)/2/2m expresses only one k. Thus, the energy E appears to be an average. It seems one may have a plane wave exp(ipx) which collides inelastically with the wall and changes to exp(-ip1x) etc.; on average energy is conserved. In (1) and (2), the spatial wavefunction W(x)= Sum over p fp exp(ipx) is obtained by solving the time-independent Schrodinger equation, and fp from taking the Fourier transform of W(x). These are strictly math procedures it seems. In a previous note (3), the time independent Schrodinger equation is written with V(x) and W(x) both expressed in terms of Fourier transforms. This lead to an equation for each p of the form: Sum j+k=p Vk fk = (E-p*p/2m) fp ((1)). In such a case, V(x) exists throughout x. For the case of an infinite wall, there is still a potential, but only at the end walls. In this note, we suggest ((1)) may be used to obtain fp without taking the Fourier transform of W(x) or solving for Vk. This approach also shows cycling is involved, even for a particle in a box. The cycling occurs at the walls.