Semiclassical oscillation of the electron through the nucleus of the H atom yields both the exact energy and the correct orbital angular momentum for l=0 quantum states. Similarly, electron oscillation through the nuclei of H2+ accounts for a stable molecule ion with energy close to the quantum mechanical solution. The small discrepancy arises from the neglect of the electron's wave nature.
Semiclassical oscillation of the electron through the nucleus of the H atom yields both the exact energy and the correct orbital angular momentum for l=0 quantum states. Similarly, electron oscillation through the nuclei of H2+ accounts for a stable molecule ion with energy close to the quantum mechanical solution. The small discrepancy arises from the neglect of the electron's wave nature.
The old quantum theory of Bohr and Sommerfeld was abandonned for the wrong reason. Its contradictions were caused not by the orbit concept but by a mental barrier—the inconceivability that an electron might collide with the atomic nucleus. Removing that barrier resolves the theory’s main failures—incorrect orbital momenta, He atom, H + 2 molecule ion. The inclusion of electron oscillations through the nucleus—a concept called “Coulomb oscillator”—renders the old quantum theory consistent with quantum mechanics (although devoid of wave character). The triple success of the Bohr-Sommerfeld model is its correct description of the H atom (and one-electron ions) concerning (1) the energy levels Enl, (2) the orbital angular momenta Lnl—if corrected as L 2 = l(l + 1)~ 2 and with the Coulomb oscillator included—and (3) the orbits’ space quantization—with (Lnl)z = ml~. These achievements are succinctly represented by the principal, angular and magnetic quantum numbers (n, l, ml) and visualized by orbital ellipse geometry—major axis, vertex curvature, and tilt angle, respectively. Orbit geometry also accounts for the average orbit size. Moreover, the Coulomb oscillator provides a natural explanation of (isotropic) hyperfine interaction. The shortcomings of the old quantum theory lie in its neglect of three properties of particles—their spin, their wave nature and their quantum statistics. These deficiencies notwithstanding, the visual appeal of the Bohr-Sommerfeld model remains a pedagogical asset to complement the abstract character of quantum mechanics.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
The problem of the motion of a particle attracted by two fixed centers of force according to the Coulomb force law can be treated by classical mechanics and has been used in quantum theory by Pauli and Messen for a theory of the hydrogen molecule ion.1 In the quantum mechanics, where the energy levels are determined as the “eigenwerte” of Schrodinger’s equation, the variables are separable and the boundary value problem is easily set up. But thus far a satisfactory treatment of the differential equations involved is lacking. Burrau2 has recently carried out a numerical integration of the problem for the lowest energy level of an electron moving under the influence of two fixed centers of Coulomb attraction as a function of the distance apart of these centers. In this paper, Burrau’s data are used to give a semi-quantitative discussion of the neutral hydrogen molecule. His values are:
Nuclear separation
1.0
1.3
1.6
1.8
2.0
2.2
2.4
2.95
Electronic energy
2.896
2.648
2.436
2.309
2.204
2.109
2.025
1.836
The unit of separation is the Bohr l1 orbit radius of hydrogen atom, that of energy is the ionization potential of atomic hydrogen.
The old quantum theory of Bohr and Sommerfeld was abandonned for the wrong reason. Its contradictions were caused not by the orbit concept but by a mental barrier--the inconceivability that an electron might collide with the atomic nucleus. Removing that barrier resolves the theory's main failures--incorrect orbital momenta, He atom, H2+ molecule ion. The inclusion of electron oscillations through the nucleus--a concept called "Coulomb oscillator"--renders the old quantum theory consistent with quantum mechanics (although devoid of wave character). The triple success of the Bohr-Sommerfeld model is its correct description of the H atom (and one-electron ions) concerning (1) the energy levels Enl, (2) the orbital angular momenta Lnl--if corrected as Lnl^2 = l(l+1) hbar^2 and with the Coulomb oscillator included--and (3) the orbits' space quantization--with (Lnl)z = ml hbar. These achievements are succinctly represented by the principal, angular and magnetic quantum numbers (n, l, ml) and visualized by orbital ellipse geometry--major axis, vertex curvature, and tilt angle, respectively. Orbit geometry also accounts for the average orbit size. Moreover, the Coulomb oscillator provides a natural explanation of (isotropic) hyperfine interaction. The shortcomings of the old quantum theory lie in its neglect of three properties of particles--their spin, their wave nature and their quantum statistics. These deficiencies notwithstanding, the visual appeal of the Bohr-Sommerfeld model remains a pedagogical asset to complement the abstract character of quantum mechanics.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.