RELATIVISTIC QUANTUM MECHANICS

Abstract

Attempts to establish a relativistic quantum mechanics-an integration of quantum theory and the theory of relativity-predate the emergence of quantum mechanics in the 1920s. Shortly after Niels Bohr had proposed his atom model in 1913, he and a few others realized that quantum theory might be improved by using relativity rather than classical mechanics. These efforts culminated in 1916-17 when Arnold Sommerfeld in Munich devised a modified version of Bohr's model by incorporating the relativistic variation of the mass of an electron moving around an atomic nucleus. That is, rather than assuming the mass to be constant, Sommerfeld adopted the expression m(v) = m 0 (1-v 2 /c 2) − 1 / 2 , where v is the elec-tron's velocity and c the velocity of light. The result was an expression of the energy levels in hydrogen-like atom Bohr's atom model that predicted a fine structure with a separation in frequency proportional to α 2 Z 2 , where α is the fine-structure constant and Z the nuclear charge (or atomic number). Sommerfeld's theory received experimental support from measurements in both the optical and the X-ray region, and the confirmation was widely seen as a triumph of the Bohr-Sommerfeld atomic model as well as the special theory of relativity. A few physicists believed that gravitation theory, in the form of Einstein's general theory of relativity, had to be incorporated in atomic theory. The Kepler motion of electrons around an atomic nucleus was analyzed by means of general relativity by Georg Jaffé, Mandoval Vallarta and others in 1922-25; however, their works were ignored by most mainstream physicists who believed that general relativity was of no importance in atomic physics. In a paper of 1922, Erwin Schrödinger applied Hermann Weyl's extension of Einsteinian general relativity to atomic theory. Although Schrödinger's paper would later come to appear as prescient, at the time his work attracted no more attention than other theories in the same tradition. Louis de Broglie's innovative theory of 1922-23, which postulated the existence of matter waves, was solidly founded on the (special) theory of relativity. According to de Broglie, quantum theory and special relativity theory were unified by the relativistic formula mc 2 = hν = hc/λ, or λ = h/p (where λ is the wavelength associated with the momentum p of some particle, whether a light quantum or an electron). In late November 1925 Schrödinger reached the decision that to transform de Broglie's hypothesis into a wave theory of atomic structure he would need a wave equation governing the behaviour of de Broglie's somewhat mysterious