Encyclopedia > Density matrix

  Article Content

Density matrix

A density matrix is used in quantum theory to describe the statistical state of a quantum system[?]. It is the quantum-mechanical analogue to a phase-space density[?] (probability distribution of position and momentum) in classical statistical mechanics. The need for a description via the density matrix arises whenever the exact quantum-mechanical state of the system (i.e. its wavefunction) is not known. Then only the probability of the system being in a certain state can be given, which is accomplished by the density matrix. In such a case, the system is said to be in a mixed state[?], while otherwise it is in a pure state[?].

Typical situations in which a density matrix is needed include: a quantum system in thermal equilibrium (at finite temperatures), nonequilibrium time-evolution that starts out of a mixed equilibrium state, and entanglement between two subsystems, where each individual system must be described by a density matrix even though the complete system may be in a pure state.

The density matrix (commonly designated by ρ) is an operator acting on the Hilbert space of the system in question. For the special case of a pure state, it is given by the projector of this state. For a mixed state, where the system is in the quantum-mechanical state |ψj⟩ with probability pj, the density matrix is the sum of the projectors, weighted with the appropriate probabilities (see bra-ket notation):

ρ = ∑j pjj⟩⟨ψj|

The density matrix is used to calculate the expectation value of any operator A of the system, averaged over the different states |ψj⟩. This is done by taking the trace of the product of ρ and A:

tr[ρ A]=∑j pj ⟨ψj|A|ψj

The probabilities pj are nonnegative and normalized (i.e. their sum gives one). For the density matrix, this means that ρ is a positive semidefinite hermitian operator (its eigenvalues are nonnegative) and the trace of ρ (the sum of its eigenvalues) is equal to one.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article

... has existed all throughout human history, and probably pre-dates the emergence of effective medicine. Quackery is still abundant today, herbal ...

This page was created in 57.1 ms