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?
References:
http://farside.ph.utexas.edu/teaching/qm/lectures/node31.html
Best Answer
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!