The breakdown of classical physics

The necessity for a departure from classical mechanics is clearly demonstrated by:

1. The anomalous stability of atoms and molecules: According to classical physics, an electron orbiting a nucleus should lose energy by emission of synchrotron radiation, and gradually spiral in towards the nucleus. Experimentally, this is not observed to happen.
2. The anomalously low specific heats of atoms and molecules: According to the equipartition theorem of classical physics, each degree of freedom of an atomic or molecular system should contribute $R/2$ to its molar specific heat, where $R$ is the ideal gas constant. In fact, only the translational and some rotational degrees of freedom seem to contribute. The vibrational degrees of freedom appear to make no contribution at all (except at high temperatures). Incidentally, this fundamental problem with classical physics was known and appreciated in the middle of the nineteenth century. Stories that physicists at the start of the twentieth century thought that classical physics explained everything, and that there was nothing left to discover, are largely apocryphal (see Feynman, Vol. I, Cha. 40).
3. The ultraviolet catastrophe: According to classical physics, the energy density of an electromagnetic field in vacuum is infinite due to a divergence of energy carried by short wave-length modes. Experimentally, there is no such divergence, and the total energy density is finite.
4. Wave-particle duality: Classical physics can deal with waves or particles. However, various experiments (e.g., light interference, the photo-electric effect, electron diffraction) show quite clearly that waves sometimes act as if they were streams of particles, and streams of particles sometimes act as if they were waves. This is completely inexplicable within the framework of classical physics.