The quasi-classical limit of quantum mechanics refers, roughly speaking, to the limit ℏ → 0. Of course, ℏ is a dimensionful constant, but in practice one studies the semi-classical regime of a given quantum theory by forming a dimensionless combination of ℏ and other parameters; this combination then re-enters the theory as if it were a dimensionless version of ℏ that can indeed be varied.
The oldest example of this procedure is Planck's radiation formula ► black body radiation; Planck's constant. Indeed, the observation of Einstein [5] and Planck [8] that in the limit ℏν/kT → 0 this formula converges to the classical equipartition law may well be the first use of the ℏ → 0 limit of quantum theory; note that Einstein put ℏν/kT → 0 by letting ν → 0 at fixed T and ℏ, whereas Planck took T → ∞ at fixed ν and ℏ.
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Primary Literature
N.N. Bogoliubov: On a new method in the theory of superconductivity. Nuov. Cim. 7, 794–805 (1958).
N. Bohr: Über die Anwendung der Quantentheorie auf den Atombau. I. Die Grundpostulaten der Quantentheorie. Z. Phys. 13, 117–165 (1923).
L. Brillouin: La mécanique ondulatoire de Schrödinger; une méthode générale de resolution par approximations successives. C.R. (Paris) 183, 24–26 (1926).
P. Ehrenfest: Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Z. Phys. 45, 455–457 (1927).
A. Einstein: Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtpunkt. Ann. Phys. 17, 132–178 (1905).
W. Heisenberg: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172–198 (1927).
H.A. Kramers: Wellenmechanik und halbzahlige Quantisierung. Z. Phys. 39, 828–840 (1926).
M. Planck: Vorlesungen Über die Theorie der Wärmestrahlung (J.A. Barth, Leipzig, 1906).
E. Schrödinger: Der stetige Übergang von der Mikro- zur Makromekanik. Die Naturwissenschaften 14, 664–668 (1926).
J.H. van Vleck: The absorption of radiation by multiply periodic orbits, and its relation to the correspondence principle and the Rayleigh-Jeans law. Part I, II. Phys. Rev. 24, 330–346, 347–365 (1924).
G. Wentzel: Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik. Z. Physik. 38, 518–529 (1926).
E.P. Wigner: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932).
Secondary Literature
M.V. Berry: Semi-classical mechanics in phase space: a study of Wigner's function. Phil. Trans. Roy. Soc. (London) 287, 237–271 (1977).
M. Brack & R.K. Bhaduri: Semiclassical Physics (Westview, Boulder, 2003).
V. Guillemin & S. Sternberg: Geometric Asymptotics (American Mathematical Society, Providence, 1977).
M.C. Gutzwiller: Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
K. Hepp: The classical limit of quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974).
V. Ivrii: Microlocal Analysis and Precise Spectral Asymptotics (Springer, New York, 1998).
N.P. Landsman: Between classical and quantum, in Handbook of the Philosophy of Science Vol. 2: Philosophy of Physics, ed. by J. Butterfield and J. Earman, pp. 417–554 (North-Holland, Elsevier, Amsterdam, 2007).
R.G. Littlejohn: The semiclassical evolution of wave packets. Phys. Rep. 138, 193–291 (1986).
A. Martinez: An Introduction to Semiclassical and Microlocal Analysis (Springer, New York, 2002).
M. Nauenberg, C. Stroud, & J. Yeazell: The classical limit of an atom. Sci. Am. June, 24–29 (1994).
R.W. Robinett: Quantum wave packet revival. Phys. Rep. 392, 1–119 (2004).
B. Simon: The classical limit of quantum partition functions. Commun. Math. Phys. 71, 247–276 (1980).
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Landsman, N.P. (2009). Quasi-Classical Limit. In: Greenberger, D., Hentschel, K., Weinert, F. (eds) Compendium of Quantum Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70626-7_182
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