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| Bohr–Sommerfeld quantization | |
|---|---|
| Name | Bohr–Sommerfeld quantization |
| Field | Quantum theory |
| Introduced | 1913–1916 |
| Creator | Niels Bohr; Arnold Sommerfeld |
Bohr–Sommerfeld quantization is an early semiclassical rule that assigns discrete values to dynamical quantities by imposing integral constraints on classical orbits. Developed in the era of Niels Bohr and Arnold Sommerfeld, it formed part of the old quantum theory and influenced later work by Werner Heisenberg, Erwin Schrödinger, and Paul Dirac. The method provided successful predictions for atomic spectra and molecular rotations while revealing tensions later resolved by matrix mechanics and wave mechanics.
The rule arose amid debates involving figures such as Max Planck, Albert Einstein, Lord Rayleigh, Rayleigh (3rd Baron), and Hendrik Lorentz about quantization after the ultraviolet catastrophe and the introduction of the quantum hypothesis. Building on Bohr model ideas, Arnold Sommerfeld extended quantization to multiply periodic systems influenced by correspondence principles advocated by Wolfgang Pauli and correspondence discussions with Paul Ehrenfest. Researchers at institutions like the University of Copenhagen, Ludwig Maximilian University of Munich, and the Kaiser Wilhelm Institute incorporated input from contemporaries including Ralph Fowler, Alfred Landé, James Franck, and Gustav Hertz. The approach guided interpretations at conferences where delegates from Royal Society, Deutsche Physikalische Gesellschaft, and academies in Paris and St. Petersburg debated quantization rules prior to the revolutionary work of Heisenberg in 1925.
The condition specifies that for each action variable J_i of a classical integral over a closed orbit, the integral ∮ p_i dq_i equals an integer multiple of Planck's constant h (or h/2π in variants), a constraint echoing the scale set by Max Planck. Sommerfeld introduced quantization of radial and angular actions to explain fine structure in hydrogen spectra studied by experimentalists such as J. J. Thomson and Johannes Rydberg. The method uses canonical variables from Hamiltonian mechanics developed by William Rowan Hamilton and integrates around tori as in work influenced later by Henri Poincaré and Joseph-Louis Lagrange. Specific formulations invoked quantum numbers analogous to those in discussions by Arnold Sommerfeld, Niels Bohr, and critiques by Wolfgang Pauli, with quantized energy levels compared to spectroscopic series cataloged by Johannes Rydberg and measured in laboratories led by Ernest Rutherford, Hans Geiger, and James Chadwick.
Sommerfeld's application to the hydrogen atom refined the Bohr model predictions to include relativistic corrections attributed to Albert Einstein's special relativity and produced fine-structure splittings matching observations from experiments linked to Gustav Kirchhoff and Joseph von Fraunhofer spectral studies. The quantization was applied to the motion of electrons in fields, cyclotron orbits studied by Ernest Lawrence, and to molecular rotation and vibration problems addressed by Linus Pauling and Max Born. It informed semiclassical analyses of systems later investigated by Isidor Isaac Rabi, Felix Bloch, Lev Landau, and Enrico Fermi, and provided approximations used in scattering contexts examined by Werner Heisenberg and Max Born in cross sections relevant to Rutherford scattering. The rule also appeared in early treatments of the Zeeman effect and Stark effect, areas explored by Pieter Zeeman and Johannes Stark respectively.
Although superseded by matrix mechanics and wave mechanics formalized by Werner Heisenberg, Erwin Schrödinger, and reconciled by Paul Dirac, the Bohr–Sommerfeld conditions anticipated semiclassical methods such as the WKB approximation developed by Gregory Wentzel, Hendrik Kramers, and Ludwig Brillouin. Connections link the action quantization to the phase-space quantization in Dirac's theory and to adiabatic invariants studied by Paul Ehrenfest and Lev Landau. The approach illuminated correspondence limits emphasized by Niels Bohr and influenced later formulations in quantum chaos research by Martin Gutzwiller and semiclassical trace formulas used by Michael Berry and Boris Chirikov. The method’s breakdown in nonintegrable systems foreshadowed advances in Hilbert space operator theory by scholars like John von Neumann.
Generalizations include Maslov indices introduced by Victor Maslov and phase-correction terms refined by Karl Popper-adjacent debates, while modern canonical quantization frameworks by Paul Dirac and geometric quantization approaches by Bertram Kostant and Jean-Marie Souriau subsume the older rules. The Einstein–Brillouin–Keller (EBK) quantization extended the condition to multidimensional tori with corrections linked to work by Albert Einstein on adiabatic invariants and by John B. Keller on boundary-phase contributions. Semiclassical path-integral formulations developed later by Richard Feynman relate classical action extremals to quantum amplitudes, connecting the Bohr–Sommerfeld perspective to methods employed in quantum field theory studies by Julian Schwinger and Richard Feynman.
Critiques came from theoreticians including Wolfgang Pauli and experimental mismatches highlighted by refinements needed for multi-electron atoms tackled by Ettore Majorana and by quantum-electrodynamical corrections later explained in the work of Sin-Itiro Tomonaga, Julian Schwinger, and Richard Feynman. The method fails for chaotic systems analyzed by Edward Lorenz and for many-body problems central to studies by Lev Landau and Yoichiro Nambu. Its heuristic status—relying on classical orbits—was challenged by foundational shifts culminating in formal operator and Hilbert-space frameworks championed at institutions such as Institute for Advanced Study and laboratories like Cavendish Laboratory and Los Alamos National Laboratory, where quantum theory’s probabilistic and nonlocal aspects were emphasized by figures including Albert Einstein, Niels Bohr, and John Bell.