Theclassical limit orcorrespondence limit is the ability of aphysical theory to approximate or "recover"classical mechanics when considered over special values of its parameters.^[1] The classical limit is used with physical theories that predict non-classical behavior.

Aheuristic postulate called thecorrespondence principle was introduced toquantum theory byNiels Bohr: in effect it states that some kind of continuity argument should apply to the classical limit of quantum systems as the value of thePlanck constant normalized by the action of these systems becomes very small. Often, this is approached through "quasi-classical" techniques (cf.WKB approximation).^[2]

More rigorously,^[3] the mathematical operation involved in classical limits is agroup contraction, approximating physical systems where the relevant action is much larger than the reduced Planck constantħ, so the "deformation parameter"ħ/S can be effectively taken to be zero (cf.Weyl quantization.) Thus typically, quantum commutators (equivalently,Moyal brackets) reduce toPoisson brackets,^[4] in agroup contraction.

Inquantum mechanics, due toWerner Heisenberg'suncertainty principle, anelectron can never be at rest; it must always have a non-zerokinetic energy, a result not found in classical mechanics. For example, if we consider something very large relative to an electron, like a baseball, the uncertainty principle predicts that it cannot really have zero kinetic energy, but the uncertainty in kinetic energy is so small that the baseball can effectively appear to be at rest, and hence it appears to obey classical mechanics. In general, if large energies and large objects (relative to the size and energy levels of an electron) are considered in quantum mechanics, the result will appear to obey classical mechanics. The typicaloccupation numbers involved are huge: a macroscopicharmonic oscillator withω = 2 Hz,m = 10 g, and maximumamplitudex₀ = 10 cm, hasS ≈ E/ω ≈mωx²
₀/2 ≈ 10⁻⁴ kg·m²/s = ħn, so thatn ≃ 10³⁰. Further seecoherent states. It is less clear, however, how the classical limit applies to chaotic systems, a field known asquantum chaos.

Quantum mechanics and classical mechanics are usually treated with entirely different formalisms: quantum theory usingHilbert space, and classical mechanics using a representation inphase space. One can bring the two into a common mathematical framework in various ways. In thephase space formulation of quantum mechanics, which is statistical in nature, logical connections between quantum mechanics and classical statistical mechanics are made, enabling natural comparisons between them, including the violations ofLiouville's theorem upon quantization.^[5][6]

In a crucial paper (1933),Paul Dirac^[7] explained how classical mechanics is anemergent phenomenon of quantum mechanics:destructive interference among paths with non-extremal macroscopic actionsS » ħ obliterate amplitude contributions in thepath integral he introduced, leaving the extremal actionS_class, thus the classical action path as the dominant contribution, an observation further elaborated byRichard Feynman in his 1942 PhD dissertation.^[8] (Further seequantum decoherence.)

One simple way to compare classical to quantum mechanics is to consider the time-evolution of theexpected position andexpected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy theEhrenfest theorem. For a one-dimensional quantum particle moving in a potential${\displaystyle V}$, the Ehrenfest theorem says^[9]

${\displaystyle m\frac{d}{dt}⟨x⟩=⟨p⟩;\frac{d}{dt}⟨p⟩=-⟨V^{\prime }(X)⟩.}$

Although the first of these equations is consistent with the classical mechanics, the second is not: If the pair${\displaystyle (⟨X⟩,⟨P⟩)}$ were to satisfy Newton's second law, the right-hand side of the second equation would have read

${\displaystyle \frac{d}{dt}⟨p⟩=-V^{\prime }\left(⟨X⟩\right)}$.

But in most cases,

${\displaystyle ⟨V^{\prime }(X)⟩\ne V^{\prime }(⟨X⟩)}$.

If for example, the potential${\displaystyle V}$ is cubic, then${\displaystyle V^{\prime }}$ is quadratic, in which case, we are talking about the distinction between${\displaystyle ⟨X^2⟩}$ and${\displaystyle ⟨X{⟩}^2}$, which differ by${\displaystyle (\mathrm{\Delta }X{)}^2}$.

An exception occurs in case when the classical equations of motion are linear, that is, when${\displaystyle V}$ is quadratic and${\displaystyle V^{\prime }}$ is linear. In that special case,${\displaystyle V^{\prime }\left(⟨X⟩\right)}$ and${\displaystyle ⟨V^{\prime }(X)⟩}$ do agree. In particular, for afree particle or aquantum harmonic oscillator, the expected position and expected momentum exactly follows solutions of Newton's equations.

For general systems, the best we can hope for is that the expected position and momentum willapproximately follow the classical trajectories. If the wave function is highly concentrated around a point${\displaystyle x_0}$, then${\displaystyle V^{\prime }\left(⟨X⟩\right)}$ and${\displaystyle ⟨V^{\prime }(X)⟩}$ will bealmost the same, since both will be approximately equal to${\displaystyle V^{\prime }(x_0)}$. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at leastfor as long as the wave function remains highly localized in position.^[10]

Now, if the initial state is very localized in position, it will be very spread out in momentum, and thus we expect that the wave function will rapidly spread out, and the connection with the classical trajectories will be lost. When the Planck constant is small, however, it is possible to have a state that is well localized inboth position and momentum. The small uncertainty in momentum ensures that the particleremains well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories for a long time.

Other familiar deformations in physics involve:

• The deformation of classical Newtonian into relativistic mechanics (special relativity), with deformation parameterv/c; the classical limit involves small speeds, sov/c → 0, and the systems appear to obey Newtonian mechanics.
• Similarly for the deformation of Newtonian gravity intogeneral relativity, with deformation parameter Schwarzschild-radius/characteristic-dimension, we find that objects once again appear to obey classical mechanics (flat space), when the mass of an object times the square of thePlanck length is much smaller than its size and the sizes of the problem addressed. SeeNewtonian limit.
• Wave optics might also be regarded as a deformation ofray optics for deformation parameterλ/a.
• Likewise,thermodynamics deforms tostatistical mechanics with deformation parameter1/N.

• Classical probability density
• Ehrenfest theorem
• Madelung equations
• Fresnel integral
• Mathematical formulation of quantum mechanics
• Quantum chaos
• Quantum decoherence
• Quantum limit
• Semiclassical physics
• Wigner–Weyl transform
• WKB approximation

• Hall, Brian C. (2013),Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer,Bibcode:2013qtm..book.....H,ISBN 978-1-4614-7115-8

• Concepts in physics
• Quantum mechanics
• Theory of relativity
• Philosophy of science
• Emergence

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