Classical limit

The classical limit or correspondence limit is the ability of a physical 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.

Quantum theory

A heuristic postulate called the correspondence principle was introduced to quantum theory by Niels Bohr: in effect it states that some kind of continuity argument should apply to the classical limit of quantum systems as the value of the Planck 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,

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

${\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ;\quad {\frac {d}{dt}}\langle p\rangle =-\left\langle V'(X)\right\rangle .}$

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

${\displaystyle {\frac {d}{dt}}\langle p\rangle =-V'\left(\left\langle X\right\rangle \right)}$.

But in most cases,

${\displaystyle \left\langle V'(X)\right\rangle \neq V'(\left\langle X\right\rangle )}$.

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

An exception occurs in case when the classical equations of motion are linear, that is, when ${\displaystyle V}$ is quadratic and ${\displaystyle V'}$ is linear. In that special case, ${\displaystyle V'\left(\left\langle X\right\rangle \right)}$ and ${\displaystyle \left\langle V'(X)\right\rangle }$ do agree. In particular, for a free particle or a quantum 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 will approximately follow the classical trajectories. If the wave function is highly concentrated around a point ${\displaystyle x_{0}}$, then ${\displaystyle V'\left(\left\langle X\right\rangle \right)}$ and ${\displaystyle \left\langle V'(X)\right\rangle }$ will be almost the same, since both will be approximately equal to ${\displaystyle V'(x_{0})}$. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for 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 in both position and momentum. The small uncertainty in momentum ensures that the particle remains 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.

Relativity and other deformations

Other familiar deformations in physics involve:

• The deformation of classical Newtonian into relativistic mechanics (special relativity), with deformation parameter v/c; the classical limit involves small speeds, so v/c → 0, and the systems appear to obey Newtonian mechanics.
• Similarly for the deformation of Newtonian gravity into
  Planck length is much smaller than its size and the sizes of the problem addressed. See Newtonian limit
  .
• Wave optics might also be regarded as a deformation of ray optics for deformation parameter λ/a.
• Likewise, thermodynamics deforms to statistical mechanics with deformation parameter 1/N.

See also

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

References