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arXiv:quant-ph/0602104

Comments on a bound state model for a two body system

L. Micu

Published Feb 13, 2006• 0 citations
arXivHTMLPDF

Abstract

We show that in classical mechanics, as well as in nonrelativistic quantum mechanics the equation of the relative motion for a two-body bound system at rest can be replaced by individual dynamical equations of the same kind as the first one, but with different parameters. We assume that in relativistic quantum mechanics the individual equations are Dirac equations with modified parameters in agreement with the individual Schrödinger equations. We find that products of solutions to the individual equations with correlated arguments are the quantum analogues of the classical representation of a bound system and represent suitable models for the bound state wave functions. As validity test for the new representation of bound states we suggest to use some observable differences between this one and the representation in terms of relative coordinates.

DOI
10.48550/arxiv.quant-ph/0602104

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We show that in classical mechanics, as well as in nonrelativistic quantum mechanics the equation of the relative motion for a two-body bound system at rest can be replaced by individual dynamical equations of the same kind as the first one, but with different parameters. We assume that in relativistic quantum mechanics the individual equations are Dirac equations with modified parameters in agreement with the individual Schrödinger equations. We find that products of solutions to the individual equations with correlated arguments are the quantum analogues of the classical representation of a bound system and represent suitable models for the bound state wave functions. As validity test for the new representation of bound states we suggest to use some observable differences between this one and the representation in terms of relative coordinates.
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In a previous note (1), we argued that both free particle quantum and classical mechanics follow from an energy flow/flux scheme based on: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) A(x,t) turns out to be the classical action tL(v) (L=Lagrangian) with x/t=v and the above hold for both the relativistic and nonrelativistic cases. We further argued that ((1a)) and ((1b)) lead to two sets of solutions, one classical and the other quantum. The classical solution assumes dt→0 and dx→0 i.e. flows may be followed at each point in space (and time). In (2) we argued that ((1a)) and ((1b)) are consistent with p=dL/dv (partial) and H=E= pv - L. From these together with p=Ev one may find: E=mo/sqrt(1-vv) (c=1). For the nonrelativistic case, E= mocc + .5mvv. In this classical analysis one may resolve each t and x, thus if one has a function V(x) (potential) representing energy, the nonrelativistic result is: (E-V(x)) = .5mvv. The quantum solution to ((1a)) and ((1b)) is very different. It establishes eigenvalue equations i.e. id/dt (partial) exp(iA) = E exp(iA) and -id/dx (partial) exp(iA) = p exp(iA) for the free particle. The importance of this is that new physical mechanisms seem to come into play. A= -Et+px so exp(-iEt+ipx) suggests time and spatial regions proportional to 1/E and 1/p in which there is some ambiguity i.e. there are two dimensions and there seems to be some kind of probability for the particle to be in one dimension or the other at x (and similar results for t) i.e. cos(px) and sin(px). As a result, one may not have dx→0 and dt→0. Furthermore, it is argued that a two dimensional associated probability scheme must accompany energy flow. At first this may seem strange, but if one considers a photon which has an energy flow Ev, one has electric and magnetic fields perpendicular to each other and perpendicular to the motion which undergo periodic motion linked to cos(wt) and sin(wt). Thus, the quantum particle periodic probability mimics an already known result from Maxwell’s equations. In fact, we have argued that for fermions there is additional angular motion i.e. spin linked with the recognition of 3-space. Thus we argue that quantum mechanics is involved with periodic type probability motions which are needed to resolve dx and dt (linked to exp(-iEt+ipx) and periodic probability motion (spin) and are also needed for fermions to resolve 3-space as two p vectors may have point in different directions (i.e. spin). Quantum mechanics therefore seems to be a very different theory from classical mechanics, but both appear to be based on energy flow/flux. As a result, the average of a quantum bound state should be linked to a classical system we argue in this note as one only deals with energy flow/flux.
In this paper I develop a relativistic Hamiltonian and Schr\"odinger equation for three particles using a generalization of the assumptions that give rise to Todorov's relativistic quasipotential equation for two particles. This formalism is treated by a method that leads to a generalization of the separable Lagrange equilateral-triangle solution from gravitational nonrelativistic to relativistic nongravitational bound-state problems in quantum as well as classical mechanics. As an example I derive an analytic solution for the spectrum of a three-body system bound by scalar Coulomb forces.
Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.
A system of identical nonrelativistic particles is considered. It is shown that the wave functions for relative motion, which have the correct permutation symmetry, must satisfy two functional equations. In the case of bosons these equations are solved for those bound states where the wave function is also in a single-product form. The only solutions are Gaussian functions. Consequently these are the only functions which can reduce the N-body energy expectation to an integral over a single variable. Furthermore, we show that our reduced two-body Hamiltonian which in general gives energy lower bounds yields the exact energy of the entire system only for the Hooke's law interaction. Neither possibility is allowed by fermions.