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Physical Review

Noninvariance Group for the Dirac Coulomb Problem

Harold Milton Kiefer, David Milton Fradkin

Published Apr 25, 1969• 7 citations
Publisher

Abstract

All bound-state solutions of the Dirac Coulomb problem are shown to provide a Hilbert space for irreducible representations of the de Sitter group $O(4, 1)$ Selected solutions of this problem provide the basis for other representations of $O(4, 1)$ These results are compared to analogous results previously obtained for the Schr\"odinger Coulomb problem. Operator forms of the generators of $O(4, 1)$ are explicitly constructed as functions of p and r which operate on the Hilbert space provided by the Dirac problem.

DOI
10.1103/physrev.180.1282

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All bound-state solutions of the Dirac Coulomb problem are shown to provide a Hilbert space for irreducible representations of the de Sitter group O(4,1)O(4, 1)O(4,1) Selected solutions of this problem provide the basis for other representations of O(4,1)O(4, 1)O(4,1) These results are compared to analogous results previously obtained for the Schr"odinger Coulomb problem. Operator forms of the generators of O(4,1)O(4, 1)O(4,1) are explicitly constructed as functions of p and r which operate on the Hilbert space provided by the Dirac problem.
With the use of a simple similarity transformation which brings the radial wave equations of the Dirac-Coulomb problem into a form nearly identical to those of the Schr\"odinger and Klein-Gordon equations, we derive simplified solutions to the Dirac-Coulomb equation for both the bound and continuum states following the familiar standard procedure adopted in the derivation of the conventional solutions. We show that to obtain the desired form of the second-order radial equations we can still work with a first-order partial differential equation rather than with the second-order Dirac equation widely employed in the derivation of the simplified solutions, and thus we can avoid the task of reducing the solutions of the second-order equations to those of the original Dirac equation. The transformed Dirac-Coulomb radial equations are so simple that one can apply the WKB method to them in the same way that one applies the WKB approximation to the Schr\"odinger radial equation, without making the further approximations commonly invoked, and the Sommerfeld-Dirac discrete spectrum follows immediately. For small Z\ensuremath{\alpha}, we also present approximate expressions for both the transformed and the original Dirac-Coulomb wave functions, which are valid for all energies and certainly valid in the quasirelativistic approximation which takes account of relativistic effects. The structure of these approximate expressions also suggests one of the methods to obtain relativistic approximate wave functions from the Pauli-spinor wave function for more general central potentials arising in atoms, molecules, and solids.
The theoretical investigation of time-dependent quantum systems requires the solution of the time-dependent Schrödinger (Dirac) equation. The basis generator method presented here allows a systematic construction of dynamically adapted wavefunctions based on a decomposition of the Hilbert space into a hierarchical structure of finite subspaces. For the class of interactions obeying an inverse integer power law, e.g., Coulomb and polarization interactions, an explicit representation of the dynamically optimized basis set is given.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations in the Lorentz-covariant world. It is possible to integrate these three papers to produce the harmonic oscillator wave function which can be Lorentz-transformed. In addition, Dirac, in 1963, considered two coupled oscillators to derive the Lie algebra for the generators of the O(3,2) de Sitter group, which has ten generators. It is proven possible to contract this group to the inhomogeneous Lorentz group with ten generators, which constitute the fundamental symmetry of quantum mechanics in Einstein’s Lorentz-covariant world.
We investigate the existence of noninvariance groups in the second-quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentum j is treated in detail. We show that the largest noninvariance group is a unitary group U(22j+1). The explicit form of its generators is given both in the m scheme and in the seniority—angular-momentum basis. The full set of 0-, 1-, 2-, ⋯, (2j + 1)-particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(22j+1). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(22j+1). These concepts are applied to the treatment of many-body forces, the state-labeling problem, and the quasiparticle picture. Finally, the generalization to more complex configurations is briefly discussed.
The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations in the Lorentz-covariant world. It is possible to integrate these three papers to produce the harmonic oscillator wave function which can be Lorentz-transformed. In addition, Dirac, in 1963, considered two coupled oscillators to derive the Lie algebra for the generators of the O(3, 2)O(3,\,2)O(3,2) de Sitter group, which has ten generators. It is proven possible to contract this group to the inhomogeneous Lorentz group with ten generators, which constitute the fundamental symmetry of quantum mechanics in Einstein's Lorentz-covariant world.