**Quantum mechanics** (QM; also known as **quantum physics** or **quantum theory**), including quantum field theory, is a branch of physics which is the fundamental theory of nature at the smallest scales of energy levels of atoms and subatomic particles. Classical physics (the physics existing before quantum mechanics) derives from quantum mechanics as an approximation valid only at large (macroscopic) scales. Quantum mechanics differs from classical physics in that: energy, momentum and other quantities are often restricted to discrete values (quantization), objects have characteristics of both particles and waves (i.e. wave-particle duality), and there are limits to the precision with which quantities can be known (uncertainty principle). Quantum mechanics gradually arose from Max Planck’s solution in 1900 to the black-body radiation problem, and Albert Einstein’s 1905 paper which offered a quantum-based theory to explain the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle.

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