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The first difference is that whereas classical theory always deals with continuously varying quantities, quantum theory must also deal with discontinuous or indivisible processes (e.g., the unit of energy packed in a quantum). The second difference is that whereas classical theory completely determines the relationship between variables at an earlier time and those at a later time, quantum laws determine only probabilities of future events in terms of given conditions in the past.
The Correspondence Principle states that the laws of quantum physics must be so chosen that in the classical limit, where many quanta are involved (e.g., n is a large integer in E=nh), the quantum laws lead to the classical equations as an average. This requirement combined with indivisibility, and incomplete determinism define the quantum theory in an almost unique manner.