On operators, superoperators, hamiltonians, and liouvillians
Per‐Olov Löwdin · International Journal of Quantum Chemistry · 2009
In the quantum theory of matter, the wavefunctions Ψ form a linear space {Ψ} which serves as a carrier space for the operators F corresponding to physical observables as well as for the system operatorsΓ associated with physical ensembles. The mappings of the operators T in turn are called superoperators, and, of particular importance are the Liouvillian and the superevolution operator used in describing the time dependence of the system operators Γ(t). The theory of such superoperators is reviewed in some detail. By studying the time dependence of expectation values (F) and time-correlation functions (F(t)G(t')), the connection between the superoperator approach and the conventional propagator method is explored. The eigenvalue problem of the superoperators is treated, and the principle importance of the superresolvents and the practical usefulness of the partitioning technique are employed. The “inner projections” of superoperators with respect to both orthogonal projectors and skew projectors are studied, and it is shown that they render “rational approximations” of the quantities involved which are always convergent. It is finally shown that the partitioning technique may be used also in solving the time-dependent Liouville equation.