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Quantum state
The quantum state vector |ψ⟩ is an abstract symbol that stands for a physical state. In this way the laws of quantum mechanics can be written as algebraic equations.
Explanation
States are expressed as numbers in a vector space. These numbers refer to the probabilities of the possible outcomes of a measurement on the system. The so-called amplitudes are complex numbers that indicate how likely each measuring outcome is.
Example 1
A photon can be polarized horizontally or vertically. Until the polarization is measured, the photon is in a superposition of these two states, and we write this as
|ψ⟩ = α |H⟩ + β |V⟩
where α and β are complex probability amplitudes. The chance that you measure a photon in state |H⟩ is |α|^{2} and that you measure it in state |V⟩ is |β|^{2}, because
|α|^{2} + |β|^{2} = 1