Rise and fall of the old quantum theory

Abstract

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 E nl , (2) the orbital angular momenta L nl -if corrected as L 2 nl = l(l + 1) 2 and with the Coulomb oscillator included-and (3) the orbits' space quantization-with (L nl ) z = m l . These achievements are succinctly represented by the principal, angular and magnetic quantum numbers (n, l, m l ) 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.

Analysis of the Jun Ishiwara’s “The universal meaning of the quantum of action”

“…The work analyzed here is one hundred years old and it take part at the apex of attempts to understand and explain the spectroscopy data [Bucher 2008, Castro 2016, Duncan 2014, Eckert 2014, Haar 1967, Jammer 1966, Mehra 1982, Mehra 2001]. Moreover, it reflects pioneering efforts to understand the relations between quantum and classical worlds in a structure where the question of classicality of nature was more evident, because the set of OQT rules itself, included explicitly the use of classical mechanics from the beginning.…”

“…His work received major notoriety, because he made the theory more robust and detailed [Jungnickel 1986], generalized the quantum law of phase integrals to apply to systems of any number of degrees of freedom, revealing the close connection of the phase integrals with the Hamilton-Jacobi theory and applying to Bohr's atom (a deep discussion about his calculations can be found on [Castro 2016]), considering elliptical orbits that allowed to deduce not just Balmer's rule but the formula for the fine structure of hydrogen spectra. It was considered the highest point of OQT [Bucher 2008, Jammer 1966], referred to by Planck as "an achievement fully comparable with that of the famous discovery of the planet Neptune whose existence and orbit was calculated by Leverrier before the human eye had seen it" [Planck 2016]. At last, is convenient to mention that the meaning of the Planck's constant has changed to Sommerfeld between 1911 and his final communications.…”

“…During that time, the physics of the atomic phenomena evolved into a theory now known as the old quantum theory (OQT) [Bucher 2008, Jammer 1966, Tomonaga 1968, which was based on a heuristic approach to atomic phenomena according to the following rules [Jammer 1966]:…”

“…[Bohr 1913a], where he uses it to obtain the correct dependence between the electronic frequencies of emission/absorption and translation around the nucleus. An improvement of the OQT would be attained through phase-space analysis, something that is not surprising, since Planck's constant h (see translations of original Planck's papers "On an improvement of Wien's equation for the spectrum" and "On the theory of the energy distribution law of the normal spectrum", with historical analysis, on [Haar 1967, Planck 2000a, Planck 2000b) has the dimension of an action (and angular momentum) [Bucher 2008, Eckert 2014, Feldens 2010, Studart 2000]. New conditions for quantization then emerged, proposed by William Wilson (1875Wilson ( -1965, Jun Ishiwara (1881Ishiwara ( -1947, Arnold Sommerfeld (1868Sommerfeld ( -1951 and Bohr.…”

On the Bohr radius relationship to spin-orbit interaction, spin magnitude, and Thomas precession

“…It is important to the current work therefore that recently the Sommerfeld model has been reinterpreted by Bucher [2] so that it provides much better agreement with modern quantum theory of hydrogenic atoms and the hydrogen molecule ion. By ruling out the circular orbit while admitting the possibility of a zero orbital angular momentum state termed by Bucher the Coulomb oscillator [3], close correspondence and agreement with modern quantum theory is obtained within the scope of applicability of the Sommerfeld model.…”