# The Density Matrix and the Liouville Equation

Define an atomic system and form its ket vector.

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Out[3]//MatrixForm= |

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Taking the outer product of the ket with its dual gives the density matrix in terms of the state vector coefficients.

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Evidently, the diagonal matrix elements of the density operator are the probabilities of the corresponding states (here the states 1 and 2). The off-diagonal matrix elements are known as

*coherences*.

We can retrace the example given in

"The State Vector and the Schrödinger Equation" in terms of the density-matrix elements rather than the state-vector amplitudes. The equation governing the evolution of the density matrix,

, known as the Liouville equation, can be found directly from the Schrödinger equation.

The function

DensityMatrix forms the density matrix for an atomic system.

Out[5]//MatrixForm= |

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The Hamiltonian describing the interaction with an electric field. We normalize the electric field to frequency units by dividing by the reduced dipole matrix element between states 1 and 2.

Out[6]//MatrixForm= |

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Out[7]//TableForm= |

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Out[8]= | |

List of the density-matrix variables.

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Solve the Liouville equation.

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The diagonal density-matrix elements are the probabilities of finding each state.

Out[11]= | |

Everything then follows identically as above. Note that averaging the state vectors, rather than the density matrices, would not produce satisfactory results. Also, the averaged density matrix can represent states of the ensemble that can not correspond to a single atomic state vector. For example, the density matrix

represents atoms that are equally likely to be in state 1 or 2, but have no well-defined phase between the amplitudes to be in these states, whereas a state vector always has a particular phase between its components.