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Old quantum mechanics by Bohr and Sommerfeld from modern perspective

Kamal Barley, Andreas L. Ruffing, Sergeĭ K. Suslov

Uspekhi Fizicheskih NaukAug 1, 20251 citation

Magnetism in matter before the discovery of quantum spin: Bohr's less well-known contribution to the transition from classical to quantum physics.

Jean-François S. Van Huele

How does one explain magnetic effects in matter when one views matter as a collection of classical charges in motion? The answer is: not at all! This is one of the points that Niels Bohr made in his doctoral dissertation in 1911, two years before addressing the issue of the stability of the hydrogen atom. The result, later rediscovered by H.J. van Leeuwen was amplified and formalized in Van Vleck’s 1932 text on electric and magnetic susceptibilities and it is currently known as the Bohr-van Leeuwen theorem. We will review Bohr’s two derivations, one statistical and one based on the motion of individual electrons. We will then propose reasons why this result, unlike that on the stability of hydrogen, did not lead to a major development in quantum theory but, instead, had to wait until after the introduction of spin and exchange forces in quantum mechanics to become generally known. The classical-quantum discontinuity • Classical Theory (mechanics, field theory, thermal, ...) 1. Critical discoveries: x, α, β, γ, e (e/m), nucleus 2. h-bar effects: Planck, Einstein, Compton, ... 3. Atomic models and stability of matter (Old quantum theory: Bohr & Sommerfeld) • Quantum Mechanics (Schrodinger, Heisenberg, Dirac, ...) Did we miss something? Classical Magnetism (1) • B fields in vacuum: Maxwell’s equation • Lorentz’ force: F = q ( E + v x B ) • Magnetism in matter susceptibilities (~T) Curie 1895 OK Zeeman effect Lorentz 1896 OK diamagnetism Langevin 1905 OK paramagnetism Langevin 1905 ? & permanent magnets Weiss 1907 ? Classical Magnetism (2) Failures... Anomalous Zeeman effect Anomalously large molecular fields ... and demise! Bohr-Van Leeuwen theorem: Lack of relevance of classical theory & Need for quantum theory!How does one explain magnetic effects in matter when one views matter as a collection of classical charges in motion? The answer is: not at all! This is one of the points that Niels Bohr made in his doctoral dissertation in 1911, two years before addressing the issue of the stability of the hydrogen atom. The result, later rediscovered by H.J. van Leeuwen was amplified and formalized in Van Vleck’s 1932 text on electric and magnetic susceptibilities and it is currently known as the Bohr-van Leeuwen theorem. We will review Bohr’s two derivations, one statistical and one based on the motion of individual electrons. We will then propose reasons why this result, unlike that on the stability of hydrogen, did not lead to a major development in quantum theory but, instead, had to wait until after the introduction of spin and exchange forces in quantum mechanics to become generally known. The classical-quantum discontinuity • Classical Theory (mechanics, field theory, thermal, ...) 1. Critical discoveries: x, α, β, γ, e (e/m), nucleus 2. h-bar effects: Planck, Einstein, Compton, ... 3. Atomic models and stability of matter (Old quantum theory: Bohr & Sommerfeld) • Quantum Mechanics (Schrodinger, Heisenberg, Dirac, ...) Did we miss something? Classical Magnetism (1) • B fields in vacuum: Maxwell’s equation • Lorentz’ force: F = q ( E + v x B ) • Magnetism in matter susceptibilities (~T) Curie 1895 OK Zeeman effect Lorentz 1896 OK diamagnetism Langevin 1905 OK paramagnetism Langevin 1905 ? & permanent magnets Weiss 1907 ? Classical Magnetism (2) Failures... Anomalous Zeeman effect Anomalously large molecular fields ... and demise! Bohr-Van Leeuwen theorem: Lack of relevance of classical theory & Need for quantum theory!

Bulletin of the American Physical SocietyMar 12, 20080 citations

Energy Levels of Harmonic Oscillator in Classical-Like Formulation of Quantum Mechanics

Vladimir I. Man’ko

Jan 1, 19971 citation

Coulomb oscillations as a remedy for the helium atom

Manfred Bucher

The largest failure of the old, Bohr-Sommerfeld quantum theory was with the helium atom. It brought about the theory's demise. I show that this failure does not originate, as commonly believed, with the orbit concept per se. Instead, it was caused by the wrong choice of orbits, compounded by ignorance of the exclusion principle. Choosing semiclassical electron oscillations through the He nucleus, I calculate a singlet ground-state energy that rivals in accuracy with quantum-mechanical results. The same method reveals Bohr's historic energy value as the forbidden triplet ground state--a result beyond the reach of quantum mechanics. At the qualitative level, the concept of Coulomb oscillations visually explains the major features in the He double spectrum in terms of crossed or parallel orbit orientation.The largest failure of the old, Bohr-Sommerfeld quantum theory was with the helium atom. It brought about the theory's demise. I show that this failure does not originate, as commonly believed, with the orbit concept per se. Instead, it was caused by the wrong choice of orbits, compounded by ignorance of the exclusion principle. Choosing semiclassical electron oscillations through the He nucleus, I calculate a singlet ground-state energy that rivals in accuracy with quantum-mechanical results. The same method reveals Bohr's historic energy value as the forbidden triplet ground state--a result beyond the reach of quantum mechanics. At the qualitative level, the concept of Coulomb oscillations visually explains the major features in the He double spectrum in terms of crossed or parallel orbit orientation.

arXiv:0705.4321May 30, 20074 citations

Semiclassical phase space distributions and scattering theory of the harmonic oscillator and hydrogen atom

Nicholas Lohr

Jan 1, 20250 citations

Dynamics of a quantum particle in double-well and periodic potentials: perturbation theory

Dmitry S. Golubev, Andrei D. Zaikin

Physics Letters ADec 1, 19891 citation

Old quantum mechanics and perturbation theory

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Old quantum mechanics and perturbation theory

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