Yes, quantum mechanics does require classical mechanics as a limiting case for its formulation. Classical mechanics, which is based on Newton's laws of motion, describes the behavior of macroscopic objects and is a well-established theory for understanding the motion of everyday objects.
Quantum mechanics, on the other hand, is a fundamental theory of physics that describes the behavior of particles at the microscopic scale, such as atoms and subatomic particles. It introduces a probabilistic description of these particles, represented by wave functions, and includes concepts such as superposition and wave-particle duality.
In the macroscopic realm, where classical mechanics applies, the behavior of objects is well described by deterministic equations of motion. However, as objects become smaller and approach the scale of atoms and particles, classical mechanics fails to provide an accurate description. Quantum mechanics becomes necessary to understand the peculiar behaviors observed at the quantum level.
The relationship between classical mechanics and quantum mechanics can be understood through the concept of correspondence principle. The correspondence principle states that the predictions of quantum mechanics must match those of classical mechanics in the appropriate limit, specifically when the objects being described are large and their behavior can be approximated classically.
In the limit of large systems or large quantum numbers, the predictions of quantum mechanics converge to the predictions of classical mechanics. This means that classical mechanics emerges from quantum mechanics as a limiting case when the systems involved are macroscopic and their quantum properties can be neglected.
So, while quantum mechanics extends our understanding beyond classical mechanics to describe the behavior of microscopic systems, it still incorporates classical mechanics as a limiting case in the appropriate macroscopic regime.