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Physical Review

Primitive Quantization in the Relativistic Two-Body Problem

John Lighton Synge

Published Jan 15, 1953• 34 citations
Publisher

Abstract

A pair of events, one for each of two particles, is regarded as a point in 8-space (${V}_{8}$). The technique of Hamilton's optical method is applied to ${V}_{8}$, giving rays and associated de Broglie waves. These waves are quantized by associating a phase-cycle with an increase of $h$ in the characteristic function, and hence quantum rules appear as resonance conditions on the de Broglie waves, when the part of ${V}_{8}$ occupied by rays is multiply connected. A Hamiltonian equation is written down, the interaction between the particles being taken to be of Coulomb form, but with ordinary distance replaced by Minkowskian separation. This leads without approximation to a simple expression for quantized proper energies of the system, and from this expression follow the correct approximate energy levels for the hydrogenic atom (with ${Z}_{\ensuremath{\alpha}}$), the mass-correction factor being included. There is no approximation based on smallness of relative velocity or ratio of masses, and mathematical complexities associated with retarded potentials are avoided by the use of the relativistic Coulomb interaction.

DOI
10.1103/physrev.89.467

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A pair of events, one for each of two particles, is regarded as a point in 8-space (V8{V}_{8}V8​). The technique of Hamilton's optical method is applied to V8{V}_{8}V8​, giving rays and associated de Broglie waves. These waves are quantized by associating a phase-cycle with an increase of hhh in the characteristic function, and hence quantum rules appear as resonance conditions on the de Broglie waves, when the part of V8{V}_{8}V8​ occupied by rays is multiply connected. A Hamiltonian equation is written down, the interaction between the particles being taken to be of Coulomb form, but with ordinary distance replaced by Minkowskian separation. This leads without approximation to a simple expression for quantized proper energies of the system, and from this expression follow the correct approximate energy levels for the hydrogenic atom (with Z\ensuremathα{Z}_{\ensuremath{\alpha}}Z\ensuremathα​), the mass-correction factor being included. There is no approximation based on smallness of relative velocity or ratio of masses, and mathematical complexities associated with retarded potentials are avoided by the use of the relativistic Coulomb interaction.
The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.
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In this centennial anniversary year of the Schrödinger equation, we present this manuscript in which we revisit the pair function describing the motions of a pair of particles in a Coulomb potential. Interest in the pair function is rekindled by recent reports that the multi-particle Hamiltonian can be written as a sum of pair Hamiltonians and coupling interactions between pairs. The coupling interactions represent correlated motions between pairs. As is shown recently, the canonical results—obtained by direct solution of the eigenequation—yield local momenta that are complex-valued. Following the suggestion of Schrödinger, we use de Broglie waves to determine eigensolutions to the Schrödinger equation that uniformly have real-valued local momenta. The angular functions obtained here resemble the canonical spherical harmonics: the Legendre polynomial is the real part of the de Broglie wave, while the equatorial function is identical to the canonical result. The radial expression for bound states has similar behaviour: the canonical function (which is the associated Laguerre function) is the real part of the corresponding de Broglie wave.
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The de Broglie relation is revisited in connection with an ab initio relativistic description of particles and waves, which is the same treatment that historically led to this famous relation. In the same context of the Minkowski four-vector formalism, we also discuss the phase and the group velocity of a matter wave, explicitly showing that both transform as ordinary velocities under a Lorentz transformation. We show that such a transformation rule is a necessary condition for the covariance of the de Broglie relation, and stress the pedagogical value of the Einstein-Minkowski-Lorentz relativistic context in the presentation of the de Broglie relation.