Eigenstate

In quantum physics, quantum state refers to the state of an isolated quantum system. A quantum state provides a probability distribution for the value of each observable, i.e. for the outcome of each possible measurement on the system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior.

A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, all other states are called mixed quantum states.

Mathematically, a pure quantum state can be represented by a ray in a Hilbert space over the complex numbers. The ray is a set of nonzero vectors differing by just a complex scalar factor; any of them can be chosen as a state vector to represent the ray and thus the state. A unit vector is usually picked, but its phase factor can be chosen freely anyway. Nevertheless, such factors are important when state vectors are added together to form a superposition.

Hilbert space is a generalization of the ordinary Euclidean space and it contains all possible pure quantum states of the given system. If this Hilbert space, by choice of representation (essentially a choice of basis corresponding to a complete set of observables), is exhibited as a function space, a Hilbert space in its own right, then the representatives are called wave functions.

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