One example of a phenomenon that cannot be fully explained by classical mechanics but can be explained by quantum mechanics is the phenomenon of quantum tunneling.
Quantum tunneling occurs when a particle passes through a potential energy barrier that it classically should not be able to overcome. According to classical mechanics, a particle with insufficient energy to overcome a barrier would be reflected back or stopped by the barrier.
However, in quantum mechanics, particles are described by wave functions that allow for the phenomenon of tunneling. The wave function of a particle extends throughout space and allows for the possibility of the particle "tunneling" through the barrier.
The probability of tunneling through a barrier can be calculated using mathematical tools such as the Schrödinger equation, which is a fundamental equation in quantum mechanics. The Schrödinger equation provides a way to describe the behavior of wave functions and determine the probabilities associated with different outcomes.
Quantum tunneling has been observed and verified in various experiments, such as in scanning tunneling microscopy and in the decay of radioactive nuclei.
This is just one example where classical mechanics fails to explain a phenomenon, but quantum mechanics, with its mathematical framework, provides a more comprehensive understanding of the underlying physics.