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Interaction with a heat bath
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Quantum mechanics in the presence of a heat bath requires the use
of density matrices. Instead of the usual von-Neumann equation
rhodot = rho \lp H
(for \lp see the section on 'Quantum-classical correspondence'),
the dynamics of the density matrix is given by a dissipative version
of it,
rhodot = rho \lp H + L(rho)
usually associated with the name of Lindblad. Here L(rho)
is a linear operator responsible for dissipation of energy to
the heat bath; it is not a simple commutator but can have
a rather complex form.
To get the Lindblad dynamics from a Hamiltonian description of
system plus bath, one uses the projection operator formalism.
The clearest treatment I know of is in
H Grabert,
Projection Operator Techniques in Nonequilibrium
Statistical Mechanics,
Springer Tracts in Modern Physics, 1982.
The final equations for the Lindblad dynamics are (5.4.48/49)
in Grabert's book.