# Is Ehrenfest theorem equivalent to Bohr's Correspondence Principle?

Ehrenfest theorem is usually dubbed as the quantum mechanical equivalent of Newton's law and Griffiths states, in the first chapter of his textbook, that Ehrenfests theorem enables us to work with expectation values in quantum mechanics as Ehrenfest theorem tells us that expectation values obey classical laws.

On the other hand, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers.

Are the two logically equivalent?

Not quite. the correspondence principle tells us that a theory should reproduce the older theory that it aims to supersede in an appropriate limit, e.g. $\hbar \rightarrow 0$ to reproduce classical behavior from quantum mechanics or $c\rightarrow \infty$ to reproduce Newtonian kinematics from special relativity.

As you can see, this is not restricted to quantum theory, but applies much more broadly. For modern physicists, it should seem obvious that a new theory needs to be able to reproduce results from the theory it supersedes in some special case or limit. However, in the early 20th century, this may not have been as intuitive: A lot of revolutionary physics was being discovered and it was unclear how to reconcile it with well-known classical physics.

On the other hand, Ehrenfest's theorem tells us that, in the limit of a many repeated experiments, the average outcome should reproduce classical laws. While this is clearly connected to the notion of a classical limit and, as wikipedia states it, provides mathematical support to the correspondence principle, I don't think it's fair to say that it is equivalent!

The previous answer is completely right but I want to add some historical comments, since from a historical point of view they have clearly different origins.

The Correspondence Principle used to be one of the two fundamental principles of Bohr's theory (one of the main formulations of quantum theory between 1913 and 1923). Originally this principle didn't only mean that quantum theory should converge into classical results for high quantum numbers; it was actually used to "recover" information from classical physics (mostly electrodynamics) into the quantum theory, and thus make some rough predictions of the intensities or polarizations of spectral lines. Other applications were the derivation of certain "selection rules". In the new theory (Quantum Mechanics), only the "classical limit" of the Correspondence Principle was retained, thus making it a much less important feature of quantum theory. Nowadays it can be considered equivalent to perform the limits $n \rightarrow \infty$ and $h \rightarrow 0$, but technically they are not exactly the same.

Bohr himself opposed this oversimplification of his principle into nothing less than a classical limit: "The requirement that the quantum theory should go over to the classical description for low modes of frequency, is not at all a principle. It is an obvious requirement for the theory." ( Helge Kragh, Niels Bohr and the Quantum Atom, Oxford University Press 2012; p. 197)

For more information about what Bohr conceived as his Correspondence Principle, see Bokulich's Three puzzles about Bohr's Correspondence Principle (PDF) and references therein.

On the other hand, Ehrenfest's theorem is a theorem purely of Quantum Mechanics. There was no equivalent of Ehrenfest's theorem in the Old Quantum Theory (that is, before 1925). It could be argued that Ehrenfest's theorem is actually another "classical limit" of the quantum theory (of the many classical limits that may exist; just to quote another example, the $N \rightarrow \infty$ of QCD, where $N$ is the number of colours of the theory.

Just for completeness, Ehrenfest's theorem should not be confused with the Adiabatic Principle (or Ehrenfest's Principle), the second fundamental principle of Bohr's theory.