(6.12) |

where the brackets denote the commutator

(6.13) |

of operators . The density operator is assumed self-adjoint, positive and of trace class. The density matrix is defined by

(6.14) |

For a pure state the density operator is written in bracket notation as

(6.15) |

and the corresponding matrix in position space is given by

The probabilistic structure is now given by the trace operation . The expectation value of an observable becomes

(6.16) |

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