Editing Correspondence principle (section)

=== The classical limit of wave mechanics ===
{{main | Classical limit}}
The outstanding success of classical mechanics in the description of natural phenomena up to the 20th century means that quantum mechanics must do as well in similar circumstances. 
{{blockquote|text=Judged by the test of experience, the laws of classical physics have brilliantly justified themselves in all processes of motion… It must therefore be laid down as an unconditionally necessary postulate, that the new mechanics … must in all these problems reach the same results as the classical mechanics.|author=Max Born, 1933<ref name="SEP"/>}}

One way to quantitatively define this concept is to require quantum mechanical theories to produce classical mechanics results as the quantum of action goes to zero, <math>\hbar \rightarrow 0</math>. This transition can be accomplished in two different ways.<ref name=Messiah_vI/>{{rp|214}}

First, the particle can be approximated by a wave packet, and the indefinite spread of the packet with time can be ignored. In 1927, [[Paul Ehrenfest]] proved his [[Ehrenfest theorem|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.<ref name=":0" />

Second, the individual particle view can be replaced with a statistical mixture of classical particles with a density matching the quantum probability density. This approach led to the concept of [[semiclassical physics]], beginning with the development of [[WKB approximation]] used in descriptions of [[quantum tunneling]] for example.<ref name=Messiah_vI/>{{rp|231}}