A Relativistic Equation for Bound-State Problems

1951/12/15 by E. E. Salpeter, Hans A. Bethe, H. A. Bethe · 2,657 citations
Physics and Astronomy · #Atomic and Molecular Physics #Bound state #Cold Atom Physics and Bose-Einstein Condensates #Coupling constant #Dirac equation #Lorentz covariance #Lorentz transformation #Mathematical physics #Meson #Nucleon #Particle physics #Physics #Quantum and Classical Electrodynamics #Quantum electrodynamics #Quantum mechanics #Wave function #Yukawa potential

paper · doi:10.1103/physrev.84.1232

published in Physical Review 84(6), 1232-1242 (American Institute of Physics)

openalex publication_date 1951/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The relativistic S-matrix formalism of Feynman is applied to the bound-state problem for two interacting Fermi-Dirac particles. The bound state is described by a wave function depending on separate times for each of the two particles. Two alternative integral equations for this wave function are derived with kernels in the form of an expansion in powers of g², the dimensionless coupling constant for the interaction. Each term in these expansions gives Lorentz-invariant equations. The validity and physical significance of these equations is discussed. In extreme nonrelativistic approximation and to lowest order in g² they reduce to the appropriate Schr"odinger equation.One of these integral equations is applied to the deuteron ground state using scalar mesons of mass \ensuremathμ with scalar coupling. For neutral mesons the Lorentz-invariant interaction is transformed into the sum of the instantaneous Yukawa interaction and a retarded correction term. The value obtained for g² differs only by a fraction proportional to (\frac\ensuremathμM)² from that obtained by using a phenomenological Yukawa potential. For a purely charged meson theory a correction term is obtained by a direct solution of the relativistic integral equation using only the first term in the expansion of the kernel. This correction is due to the fact that a nucleon can emit, or absorb, positive and negative mesons only alternately. The constant g² is increased by a fraction of 1.1(\frac\ensuremathμM) or 15 percent.

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