Density matrix

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Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
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In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems.^[1] It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states.^{[2]: 73 [3]: 100} These arise in quantum mechanics in two different situations:

1. when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of possible preparations, and
2. when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.

Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as quantum statistical mechanics, open quantum systems and quantum information.