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In classical mechanics, electrons would be able to absorb or lose any amount of energy, which is incorrect. In quantum mechanics, electrons have both wave properties and particle properties. They are only able to absorb or lose energy in discrete amounts in order to move between energy levels. Discrete packets of energy come in the form of photons.
Classical mechanics generally pertains to larger objects, where events are more continuous and predictable. Quantum mechanics describes things on a smaller, particle level, where random transitions occur. For instance, classical mechanics describes light as a wave and quantum mechanics describes light as a particle.
I think classical mechanics describes the objects that are have a continuous movement--movements that are predictable, like a trajectory. But quantum mechanics study tiny particles like electrons, photons that we talked about in class. Theses particles are usually moving in unpredictable paths: say, an electron could be at point A right now, but then suddenly jumps to point B. So these particles have a discontinuous path which classical mechanics does not apply.
To echo some of the above responses, quantum mechanics deals in very extreme values (for example, very large numbers for speed/frequency and very low numbers for mass, momentum, wavelength, energy). In quantum mechanics, because the subjects we are calculating for are so small, there is also a certain degree of uncertainty with calculations, whereas classical mechanics deals in subjects with qualities that are more easily measurable. Ultimately though the difference is due to the unique properties of quantum subjects as waves and particles.
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