In 1906, he was also the first to write that quantum theory should replicate classical mechanics at some limit, particularly if the Planck constant h were taken to be infinitesimal.
First Sommerfeld and Max Born developed a "quantization procedure" based on the action angle variables of classical Hamiltonian mechanics.
"[9]: 138 Bohr's first article containing the definition of the correspondence principle[10]: 29 was in 1923, in a summary paper entitled (in the English translation) "On the application of quantum theory to atomic structure".
[11]: 22 In modern terms, this condition is a selection rule, saying that a given quantum jump is possible if and only if a particular type of motion exists in the corresponding classical model.
[11]: 23 Similarly he shows a relationship for the intensities of spectral lines and thus the rates at which quantum jumps occur.
[2] The PhD dissertation of Hans Kramers working in Bohr's group in Copenhagen applied Bohr's correspondence principle to account for all of the known facts of the spectroscopic Stark effect, including some spectral components not known at the time of Kramers work.
Other physicists picked up the concept, including work by John Van Vleck and by Kramers and Heisenberg on dispersion theory.
Theoretical calculations by Van Vleck and by Kramers of the ionization potential of Helium disagreed significantly with experimental values.
[9]: 175 Bohr, Kramers, and John C. Slater responded with a new theoretical approach now called the BKS theory based on the correspondence principle but disavowing conservation of energy.
Einstein and Wolfgang Pauli criticized the new approach, and the Bothe–Geiger coincidence experiment showed that energy was conserved in quantum collisions.
First, Heisenberg's 1925 Umdeutung paper on matrix mechanics was inspired by the correspondence principle, although he did not cite Bohr.
[2] Further development in collaboration with Pascual Jordan and Max Born resulted in a mathematical model without connection to the principle.
Dirac developed these connections by studying the work of Heisenberg and Kramers on dispersion, work that was directly built on Bohr's correspondence principle; the Dirac approach provides a mathematically sound path towards Bohr's goal of a connection between classical and quantum mechanics.
In 1927, Paul Ehrenfest proved his namesake theorem that showed that Newton's laws of motion hold on average in quantum mechanics: the quantum statistical expectation value of the position and momentum obey Newton's laws.
This approach led to the concept of semiclassical physics, beginning with the development of WKB approximation used in descriptions of quantum tunneling for example.
Rather than a principle, "there may be in some situations an approximate correspondence between classical and quantum concepts," physicist Asher Peres put it.
For example, classical systems can exhibit chaotic orbits which diverge but quantum states are unitary and maintain a fixed overlap.