Liouville space notation

The operators which carry the superscript L are written in Liouville-space notation. The most important simplification arising from this notation is that a transformation OpO in Hilbert space can be written as O p in Liouville space. Here exp i tr 1 describes the phase evolution in the applied gradients, and exp -iHJ (r) ti describes the evolution under the internal Hamiltonians of the sample, corresponding to the free-induction signal. Relaxation is neglected in (8.3.4). 

The evolution of the density matrix in time requires the solution of the equation of motion, Eq. (10) or (20) in a basis set of N states, this represents a system of coupled linear differential equations for the individual density matrix elements. It is most natural to consider the equation in Liouville space, where ordinary operators (N x N matrices) are treated as vectors (of length N ) and superoperators such as and which act on operators to create new operators, become simple matrices (of size X N ). In the Liouville space notation, Eq. (9) would... 

In (7.90) a slightly modified notation is introduced for convenience for the bra and ket vectors in the Liouville space for the resolvent superoperator... 

Because T operates on each element of a matrix it is called a superoperator. In fact, the Hilbert-space formulation of quantum mechanics leading to the von Neumann equation of motion of the density matrix can be simplified considerably by introduction of a superoperator notation in the so-called Liouville space. Furthermore, for the analysis of NMR experiments with complicated pulse sequences it is of great help to expand the density matrix into products of operators, where each product operator exhibits characteristic transformation properties under rotation [Eml]. 

A particularly convenient notation for trajectory bundle system can be introduced by using the classical Liouville equation which describes an ensemble of Hamiltonian trajectories by a phase space density / = f q, q, t). In textbooks of classical mechanics, e.g. [12], it is shown that Liouville s equation... 

While this compact notation is convenient for the experienced user, to grasp the basic concepts a rough description may be preferable. Therefore, for the beginners that is not familiar with the concepts of statistical mechanics we reiterate the presentation of the abstract theory using an alternative notation equivalent to what is common in the fluid dynamic literature [40] [61]. The purpose is to provide introductory ideas about the practical implications of the Liouville equations describing the ensemble flow in P-space. 

U thus solves the time dependent Schrddinger equation for the wavefunction 4 (f). The dependence of 4< on spin and space coordinates has not been retained explicitly in the notation. U solves furthermore the time dependent Liouville-von Neumann equation for the density operator Q t) in equation (22) ... 

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