The statement that "for large values of quantum numbers, classical mechanics approaches quantum mechanics" is generally false. In classical mechanics, which is the branch of physics that describes the motion of macroscopic objects, particles are treated as classical point-like objects with definite positions and momenta. On the other hand, in quantum mechanics, particles are described by wave functions that exhibit wave-particle duality and can be in superposition states.
Quantum mechanics and classical mechanics have different mathematical formalisms and describe the behavior of particles in distinct ways. While classical mechanics is deterministic and operates with well-defined trajectories, quantum mechanics is probabilistic and describes the behavior of particles through wave functions and operators.
As the quantum numbers associated with a system increase, it typically indicates a higher energy state or a larger quantum state space. In such cases, the behavior of the system is generally more accurately described by quantum mechanics rather than classical mechanics. Quantum effects become increasingly prominent and crucial in these high-energy or high-quantum-number regimes.
However, in certain circumstances, when quantum systems become large and complex, they can exhibit classical-like behavior through a process known as quantum decoherence. In such situations, the quantum system loses its quantum coherence and behaves as if it follows classical laws. This can lead to the emergence of classical behavior from quantum systems, but it is a specific case and not a general rule.
In summary, for large values of quantum numbers, classical mechanics does not approach quantum mechanics. Instead, quantum mechanics remains essential for describing the behavior of particles at the quantum level, while classical mechanics is applicable for macroscopic objects in the classical regime.