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Mechanics edit
The physics and mathematics of mechanics describes how matter interacts through forces and motion. For extremely low mass matter, quantum mechanics is needed. A system in quantum mechanics is described using a quantum state. A state gives a complex number amplitude throughout space. The magnitude of the complex number relates to the probability of detecting a particle. The phase of the number plays an important role in explaining many quantum concepts. A quantum state obeys a wave equation. For example if the motion significantly slower than the speed of light, the state obeys Schrodinger equation.
Formula for quantum states have been found for single particles, like an electron in a Hydrogen atom or in a simple theoretical potential well. Numerical calculations can be done for multiple particles but qualitative understanding relies on two important ways building up approximate quantum states.
Superposition edit
The single-particle wave equations have a set of solutions called eigenfunctions, each corresponding to a set of energy values. The sum of any of these solutions is also a solution.