Variational Principle for the Many Body Density Matrix

The Gibbs-Delbruck variational principle for the
free energy based on a trial density matrix

(121) |

(122) |

as a functional of

(124) |

(125) |

(126) |

or, using the symmetry of the density matrix in and ,

Finally, we may consider a variation at some arbitrary, fixed to get

It should be noted that in going from Eq. 3.9 to Eq. 3.10 a density matrix symmetric in and is assumed, which is a property of the exact density matrix. If the variational ansatz does not manifestly have this invariance Eq. 3.11 minimizes the quantity,

(130) |

so

In the imaginary time derivative , only variations in and not are considered since is fixed so,

(133) |

This is the imaginary-time equivalent to the approach of Singer and Smith (1986) for an approximate solution of the time dependent Schödinger equation using wave packets (see section 3.3). Introducing the notation

and using Eq. 3.14, the fundamental set of first order differential equations for the dynamics of the variation parameters in imaginary time follows from Eq.. 3.16 as,

where

and the norm matrix

with

The initial conditions follow from the free particle limit of the density matrix at high temperature, ,

Various ansatz forms for may now be used with this approach. After considering the analogy to real time wave packet molecular dynamics, the principle is first applied to the problem of a particle in an external field.