The Bohm Aharonov phase shift for a free quantum particle with momentum p moving through a magnetic potential A(x) with no magnetic force is well known as exp(i Integral dx1 (a,x) e/cA(x1)dx1. In the literature this has been obtained in two very different ways. The first way (standard) (1) is to use the classical equation for a charged particle in an electromagnetic field with electrostatic potential i.e. (p-e/cA)(p-e/cA)/2m = E with p=-id/dx representing an overall momentum, not just that of the particle. In such a case, there is coupling at the momentum level, but E is the energy of a free particle with momentum p1=p-e/cA (classically) i.e. p1p1/2m. In other words, momentum considerations are treated differently from energy ones. The time dependent wavefunction is exp(-iEt) where E shows only a noninteracting free particle. The second approach (2) is the opposite. It ignores the Schrodinger equation and the momentum picture, but retains the idea that for a nonacclerating free particle p/m = velocity=dx/dt. It then considers an interaction which changes E namely delta E = p e/cA(x)/m. This leads to Integral pA(x)/m dt = Integral pA/m dx/v which yields the same Bohm Aharonov phase shift. In a previous note (3) we considered a free particle classical action as A=-Et+px. We argued that Et and px should be kept separate leading to exp(-iEt) and exp(ipx) (not combined) if one creates an eigenvalue equation. We argued that scaling exists in both time (by E) and space (by p). At the same time, the particle moves classically with a speed v=p/m, but has an associated exp(ipx) momentum wave (and independently an energy wave exp(-iEt)). Even if one uses v=dx/dt as in (2) these waves are different i.e. Et= Integral E dt = Integral .5pv dx/v = .5 px while px is the phase from the momentum consideration. Thus one might argue that it is unusual that a change in E i.e. delta E = p/m A(x) from (2) applied to exp(-iEt) yields the same Bohm Aharonov phase as the Schrodinger equation with p, A coupling, but E left untouched In this note, we consider the classical action A= -Et+px and try to explain this feature directly from this expression.
The Bohm Aharonov phase shift for a free quantum particle with momentum p moving through a magnetic potential A(x) with no magnetic force is well known as exp(i Integral dx1 (a,x) e/cA(x1)dx1. In the literature this has been obtained in two very different ways. The first way (standard) (1) is to use the classical equation for a charged particle in an electromagnetic field with electrostatic potential i.e. (p-e/cA)(p-e/cA)/2m = E with p=-id/dx representing an overall momentum, not just that of the particle. In such a case, there is coupling at the momentum level, but E is the energy of a free particle with momentum p1=p-e/cA (classically) i.e. p1p1/2m. In other words, momentum considerations are treated differently from energy ones. The time dependent wavefunction is exp(-iEt) where E shows only a noninteracting free particle. The second approach (2) is the opposite. It ignores the Schrodinger equation and the momentum picture, but retains the idea that for a nonacclerating free particle p/m = velocity=dx/dt. It then considers an interaction which changes E namely delta E = p e/cA(x)/m. This leads to Integral pA(x)/m dt = Integral pA/m dx/v which yields the same Bohm Aharonov phase shift. In a previous note (3) we considered a free particle classical action as A=-Et+px. We argued that Et and px should be kept separate leading to exp(-iEt) and exp(ipx) (not combined) if one creates an eigenvalue equation. We argued that scaling exists in both time (by E) and space (by p). At the same time, the particle moves classically with a speed v=p/m, but has an associated exp(ipx) momentum wave (and independently an energy wave exp(-iEt)). Even if one uses v=dx/dt as in (2) these waves are different i.e. Et= Integral E dt = Integral .5pv dx/v = .5 px while px is the phase from the momentum consideration. Thus one might argue that it is unusual that a change in E i.e. delta E = p/m A(x) from (2) applied to exp(-iEt) yields the same Bohm Aharonov phase as the Schrodinger equation with p, A coupling, but E left untouched In this note, we consider the classical action A= -Et+px and try to explain this feature directly from this expression.
In this note, we consider the Hamiltonian 1/2m (p-e/c A)(p-e/c A) W = E W ((1)) in the situation where there is no magnetic field even though A(x) still exists. Thus, if there is no force, one might expect to have a free particle which satisfies -1/2m d/dx d/dx W = E W and yields a free particle spatial density. Thus, one might guess the solution to ((1)) should be a free particle wavefunction exp(ikx) with perhaps a phase added. Such a phase does not affect spatial density W*W. One may solve ((1)) using the assumption that d/dx A(x) = 0 (used in the literature (1)) to find: W= Wfree(x) exp(ie/hbar c Integral x1=0 to x A(x1)dx1). Thus, one treats this phase as if one had exp(i K x) where K=A(x). A second derivative d/dx brings down i A(x) again. In previous notes, we suggested that a potential is stochastic and may be written as: V(x)= Sum over k V(k) exp(ikx). If one expands the terms in ((1)) and uses d/dx A =0, one obtains a momentum dependent potential. This potential should be physical, we argue, because it applies to cases where there is a magnetic field as well as to the case here where there is not. Thus, the potential should give hits of the form V(k) exp(ikx) as well as couple to d/dx W. In other words, we argue these hits are real whether or not average force is zero. The effect of such hits is to create a momentum distribution at each x: P(p/x) = a(p) exp(ipx) / W(x). There is forward and backward motion, but the average kinetic energy(x) + V(x)= E at each point where: kinetic energy(x) = [Sum over p p*p/2m a(p) exp(ipx)]/W(x). Given that average kinetic energy should be p*p/2m for a free particle (if the potential yields no hits), one may consider a phase exp(i K(x) x) multiplied by Wfree(x) i.e. exp(ipx) . This seems to imply a shifted velocity i.e. a local collective velocity of K(x) where K(x) happens to equal A(x) in this case. Matters, however, seem to be more complicated as we consider a microscopic coupling of exp(ikx) terms from both Wfree(x) and exp(i Bohm Aharonov) phase just as there may be a coupling between W=WoW1 where Wo(x) is the ground state and W1(x) represents the effects of an interacting substance (e.g. a photon or phonon).
We take as our starting point the free particle classical action which may be written as A= -Et+px if t and x are treated as independent variables. It is possible to find two relationships linking v (velocity) to x and t from dA=0. The first is the classical wave result vp = E while the second is p=Ev where E=mo/sqrt(1-vv) (c=1). Furthermore one may write dA/dx partial = p ((1a)) and dA/dt partial = -E. ((1b)) This may be generalized to two eigenvalue differential equations, but this immediately begs the question: Why do this? We argue that Lorentz invariant form -Et+px represents scaling of both t and x. For example, p scales x and we argue there should be physical relevance to this. This physical relevance results in the wavelength proportional to 1/p. In other words, -Et+px represents more than velocity. The treatment of the scaling for a quantum wave and classical wave differ, we argue, in that the former keeps E,t and p,x separate while a classical wave does not. In other words pv=E applies to a classical wave which is described by cos(-Et+px). In other words, one takes ((1a)) and ((1b)) which separate E,t and p,x into two independent equations and takes account of scaling by combining the two (to include both x and t) i.e. creating the classical wave equation: 1/vv d/dt d/dt partial W(x,t) = d/dx d/dx partial W(x,t). We argue that quantum mechanics on the other hand keeps ((1a)) and ((1b)) separate. The velocity (relativistic) p=Ev is not a wave velocity. A solution to ((1a)) is exp(ipx) and to ((1b)) exp(-iEt). These are usually combined for a free particle to give exp(-iEt+ipx) and a wave is described with a phase velocity pv(phase)=E. We prefer to keep exp(-iEt) and exp(ipx) separate . For example, for two slit interference one does not make use of phase velocity or exp(-iEt). For a bound state one uses p=Ev, but a single free particle is represented by exp(ipx) in W(x)=wavefunction = Sum over p a(p)exp(ipx). exp(-iEn t) appears after one has found En, using the exp(ipx)’s in the sense that one does not include an exp(- i pp/2m t) for each exp(ipx). We argue that by separating E,t and p,x, the latter leads to a scheme for which there is “no time”. It is true that the particle moves at a speed x/t, but p=m1v1=m2v2 etc so p does not uniquely define velocity, rather it defines an impulse hit 2p = Integral F dt . Thus the scaling px which leads to an x, p only wave exp(ipx) is linked to an interaction. If the particle is moving freely and not interacting, then one has classical determinism i.e. v=constant = x/t. As soon as an interaction occurs, one changes from variables x,t which are linked to x, P conditional (x/p) and there is uncertainty in the position of x within a wavelength. The phase in exp(ipx) is only dependent on x not on the time the particle reaches x. exp(ipx) thus represents a probability and one must consider a sum of different possible outcomes (as in a two slit experiment) in order to describe an interaction. This is very different from the local interaction of classical mechanics. Nevertheless for a bound state one makes use of the two formalisms i.e. KE(x)= Sum over p pp/2m a(p)exp(ipx) / W(x) = KEclassical(x). We try to discuss some of these ideas in this note.
In a previous note (1), we argued that a potential V(x) may be written as Sum over k V(k) exp(ikx). Thus, -dV/dx is only an average force which may in fact be zero, yet -d/dx V(k) exp(ikx) is not zero leading to possible forward and backward motion/acceleration (due to each V(k) exp(ikx) which act at a different times). This gives rise to a momentum distribution P(p/x) =a(p) exp(ikx)/W(x) (W(x)=wavefunction) at each x. In this note, we wish to link these ideas to the Bohm Aharonov effect by considering two recent (2018, 2020) examples from the literature. In the first (2), the magnetic potential A(r) is written in a Fourier series in the same manner as V(k)exp(ikx). Thus, although grad A = B (magnetic field) = 0, grad A(k) exp(ikx) should not be zero and so there may be hits from “virtual photons”. (2) applies perturbation theory to find a correction to energy E and then considers exp(i Integral over t delta E dt) as in adiabatic theory. The second approach (3) applies the usual Hamiltonian containing the magnetic potential A, but imagines a little box at various xi positions such that A(xi) is a constant at each xi. In such a case, one may show directly that W(x)= W(x=0) exp( ie/hbar Integral xo to x1 A(x) dx) which yields the Ehrenberg–Siday–Aharonov–Bohm effect. In this approach, one essentially has a different constant potential at each point and hence a different energy. In a previous note, we argued that even though average kinetic energy may remain the same, a change in constant potential using Sum over k V(k) exp(ikx) gives rise to different physical effects which try to argue are related to the Bohm Aharonov effect. We also try to consider a more general case.
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 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.
In a previous note (1) we argued that the free particle classical action A (relativistic or nonrelativistic) may be written in terms of v=x/t and then varied in x and t independently. One finds that dA/dx partial = p and dA/dt partial = -E implying A= -Et+px with x and t independent. This is the opposite of the classical mechanical idea of x(t). Creating an eigenvalue equation yields -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = Eexp(iA). exp(iA) plays the role of an unusual type of probability we argue which describes two dimensional periodicity (cos(px) and sin(px) ). We have argued that the quantum particle is the medium for a two dimensional wavelike motion of an energy-momentum wave. A wave type structure is needed to describe dynamics (like a classical wave), but unlike a classical wave two dimensions are required because there is no preference of any x point for a free particle. Both cos(px) and sin(px) both treat x in a preferential manner as their values vary with x. For a classical wave this is fine because the creation point of the wave imposes boundary conditions, but there is no such boundary condition imposed on a free particle. exp(ipx-iEt0 must somehow then link to P(x)=constant. One may note that exp(-iA)exp(iA) = 1 for the free particle. This may be argued to be linked to spatial invariance i.e. P(x)=constant where P(x) is not exp(ipx). Here we argue that exp(-iA)exp(iA) represents a product of two unusual probabilities , one for motion in a certain direction and the other for the time reversed scenario. If the quantum particle is the medium of an energy-momentum wave exp(-iEt+ipx), then the product of the regular and time reversed unusual probabilities cancel or undo each other. In other words one does not see any change in space as long as one has a noninteracting exp(-iEt+ipx) . Given that -id/dx is a linear operator this product yields a momentum of 0. We consider this idea and its implications on a momentum distribution Sum over p a(p)exp(ipx) which may represent a bound state or other states. In particular, it seems that P(x)=W*(x,t)W(x,t) which yields a classical type of probability (P(x) real, positive and normalized to 1) is linked to a product (merging) of forward and time reversed situations so that any spatial density which remain due to the quantum particle acting as a medium for an energy-momentum wave (as well as the time reversed part) would represent an actual change in spatial density. exp(ipx) already contains two periodic spatial functions (cos(px) and sin(px)), but does not manifest it unless there is an interaction.