The term was coined by Max Planck, and derives from the observation that some physical quantities can be changed only by discrete amounts, or quanta, as multiples of the Planck constant, rather than being capable of varying continuously or by any arbitrary amount. For example, the angular momentum, or more generally the action,[citation needed] of an electron bound into an atom or molecule is quantized. Although an unbound electron does not exhibit quantized energy levels, one which is bound in an atomic orbital has quantized values of angular momentum. In the context of quantum mechanics, the wave–particle duality of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons and other atomic-scale objects.
The mathematical formulations of quantum mechanics are abstract. Similarly, the implications are often counter-intuitive in terms of classical physics. The centerpiece of the mathematical formulation is the wavefunction (defined by Schrödinger's wave equation), which describes the probability amplitude of the position and momentum of a particle. Mathematical manipulations of the wavefunction usually involve the bra-ket notation, which requires an understanding of complex numbers and linear functionals. The wavefunction treats the object as a quantum harmonic oscillator and the mathematics is akin to that of acoustic resonance.
Many of the results of quantum mechanics do not have models that are easily visualized in terms of classical mechanics; for instance, the ground state in the quantum mechanical model is a non-zero energy state that is the lowest permitted energy state of a system, rather than a traditional classical system that is thought of as simply being at rest with zero kinetic energy.
Fundamentally, it attempts to explain the peculiar behaviour of matter and energy at the subatomic level—an attempt which has produced more accurate results than classical physics in predicting how individual particles behave. But many unexplained anomalies remain.