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.
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.
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.
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.
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.
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.
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.
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).