The principles of quantum mechanics are broad: states of a physical system form a complex vector space and physical observables are identified with Hermitian operators that act on this Hilbert space. The correspondence principle is one of the tools available to physicists for selecting quantum theories corresponding to reality. Once the Schrödinger equation was given a probabilistic interpretation, Ehrenfest showed that Newton's laws hold on average: the quantum statistical expectation value of the position and momentum obey Newton's laws.

In the Schrödinger approach classical behavior is not clear because the waves spread out as they move. In matrix mechanics, the correspondence principle was built in and was used to construct the theory. The post-1925 new quantum theory came in two different formulations. 'Restricted QCC' refers to the first two moments of the probability distribution and is true even when the wave packets diffract, while 'detailed QCC' requires smooth potentials which vary over scales much larger than the wavelength, which is what Bohr considered. A more elaborated analysis of quantum-classical correspondence (QCC) in wavepacket spreading leads to the distinction between robust 'restricted QCC' and fragile 'detailed QCC'. Bohr provided a rough prescription for the correspondence limit: it occurs when the quantum numbers describing the system are large. The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the classical limit. Arnold Sommerfeld referred to the principle as 'Bohrs Zauberstab' (Bohr's magic wand) in 1921. Bohr's correspondence principle demands that classical physics and quantum physics give the same answer when the systems become large. If quantum mechanics were to be applicable to macroscopic objects, there must be some limit in which quantum mechanics reduces to classical mechanics.

But macroscopic systems, like springs and capacitors, are accurately described by classical theories like classical mechanics and classical electrodynamics. The rules of quantum mechanics are highly successful in describing microscopic objects, atoms and elementary particles. This concept is somewhat different from the requirement of a formal limit under which the new theory reduces to the older, thanks to the existence of a deformation parameter.Ĭlassical quantities appear in quantum mechanics in the form of expected values of observables, and as such the Ehrenfest theorem (which predicts the time evolution of the expected values) lends support to the correspondence principle. The term codifies the idea that a new theory should reproduce under some conditions the results of older well-established theories in those domains where the old theories work. The principle was formulated by Niels Bohr in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom.

In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations. In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers.

Correspondence principle definition