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For quantum statistical mechanics the Schrödinger
equation gives rise to
the von Neumann equation for the density operator
![$\displaystyle \imath \hbar \frac{\partial \rho}{\partial t} = [ H, \rho ] ,$](img338.png) |
(6.12) |
where the brackets denote the commutator
![$\displaystyle [A,B] = AB - BA$](img339.png) |
(6.13) |
of operators
.
The density operator
is assumed self-adjoint, positive
and of trace class.
The density matrix
is defined by
![$\displaystyle \rho(x_1,x_2) = \langle x_1 \vert \rho \vert x_2 \rangle .$](img342.png) |
(6.14) |
For a pure state
the density operator is
written in bracket notation as
![$\displaystyle \vert\psi\rangle\langle\psi\vert$](img343.png) |
(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
![$\displaystyle \bar{A} = {\mathrm{tr} (}A \rho) = {\mathrm{tr} (}\rho A) .$](img346.png) |
(6.16) |
Previous: 6.1.1.1 Schrödinger Equation
Up: 6.1.1 Wave Mechanics
Next: 6.1.2 Hydrodynamical Formulation
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