Systematic comparison of the quantization rules of hydrogenoid atoms in the Old Quantum, Schrödinger, Klein–Gordon, and Dirac theories, by means of a common set of three parameters
A set of three parameters λ, β, and δ related to energy E and angular momentum L were used to make the closest possible formal comparison between the five successive theories of the quantization of the energy levels of the hydrogenoid atoms. The expression of the quantization rules was made uniform and the differences between the theories were reduced to simple redefinitions of the parameters.
A set of three parameters λ, β, and δ related to energy E and angular momentum L were used to make the closest possible formal comparison between the five successive theories of the quantization of the energy levels of the hydrogenoid atoms. The expression of the quantization rules was made uniform and the differences between the theories were reduced to simple redefinitions of the parameters.
In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom, which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized,” meaning that only certain sizes, shapes, and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2)~. The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra. The new quantum theory appeared to be more revolutionary and more powerful. It was triggered by Heisenberg’s discovery of matrix mechanics and Schro¨dinger’s discovery of wave mechanics. These two formalisms would not only predict the quantization of energy and provide a prescription to determine the energy difference between the levels (and thus explain the multiplicity of atomic spectra), but also allow one to calculate easily probabilities of transitions between different quantized energy states. At first, matrix mechanics and wave mechanics looked entirely different in their mathematical appearance and physical meaning. However, Schro¨dinger and Eckart [1] independently showed that the two theories are mathematically equivalent. Toward the end of 1926, Dirac unified the two theories using the concept of state vector and thus established the transformation theory of quantum mechanics. This ultimately had a profound implication for the quantum mechanical (mathematical) recipe to treat spin, as we will show in this chapter. The transformation theory is the mathematical recipe to handle modern quantum mechanics. In Heisenberg’s matrix mechanics, a physical quantity is expressed by a matrix, whereas in Schro¨dinger’s wave mechanics, a physical quantity is expressed by a linear operator. In the unified transformation theory, physical quantities are represented by abstract linear operators called Dirac’s q-numbers, which are linear operators in an infinite-dimensional linear space. Depending upon which types of orthogonal coordinate systems are used in this linear space, either matrix mechanics or wave mechanics emerges.
In a recent paper [J. Math. Phys. 35, 3333 (1994)] Good postulated new rules of quantization, one of the major features of which is that the quantum evolution of the wave function is always given by ordinary differential equations. In this paper we analyze the proposal in some detail and discuss its viability and its relationship with the standard quantum theory. As a byproduct, a simple derivation of the ‘‘mass spectrum’’ for the Klein–Gordon field is presented, but it is also shown that there is a complete additional spectrum of negative ‘‘masses.’’ Finally, two major reasons are presented against the viability of this alternative proposal: (a) It does not lead to the correct energy spectrum for the hydrogen atom. (b) For field models, the standard quantum theory cannot be recovered from this alternative description.
In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized”, meaning that only certain sizes, shapes and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2). The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra.
Abstract The development of the complex of assumptions and methods now referred to as the “old quantum theory” mainly took place in the first five years following the introduction of the Bohr atomic model in 1913. Three guiding principles emerged that were used, sometimes in overlapping ways, to explain the flood of spectroscopic data that needed to be explained. First, quantization rules (or conditions) were proposed to single out the allowed orbital motions of electrons in atoms. These rules were derived in various forms by Planck, Sommerfeld, and Wilson, but were put into their most general form by Schwarzschild, who recognized the underlying principle as the quantization of the action variables of a multiply periodic classical system. Second, the special role of the action variables in quantization was given convincing support by the transfer of the adiabatic principle of mechanics to quantum theory (work primarily due to Paul Ehrenfest). Third, the correspondence principle, or statement of asymptotic coincidence of quantum and classical theory in the limit of large quantum numbers, originally introduced by Bohr in 1913 as a supporting argument for his quantization of angular momentum in his theory of the hydrogen atom, was extended by Bohr and Kramers to provide selection rules and approximate intensity predictions evening the regime low quantum numbers.
The energy structures and transition energies of single-electron atoms and ions are presented. Five Nobel Prizes in Physics were awarded for the theories discussed in this chapter. We first review the Bohr model, which was based on quantized angular momentum and classical circular orbits. The wave model of Schrödinger followed, in which spherical boundary conditions quantized polar and azimuthal standing waves. The energies were identical to Bohr’s, but transition selection rules dictated the change in angular momentum of the system during absorption and emission. Dirac incorporated electron spin and relativistic energies, resulting in energy shifts and fine structure splitting of the energy levels for non-zero angular momentum states. Feynman and Swinger incorporated quantization of the electric vector potential. This physics broke energy degeneracies in the Dirac model and correctly predicted the famous Lamb shift. In this chapter, each of these models are described in detail. The final full characterization of the energy levels and transitions are presented. The chapter ends with a discussion on isotope shifting and transitions to the continuum (ionization/recombination).