Liouville Hermitian

The density operator r(<) is a Hermitian and positive function of time, and satisfies the generalized Liouville-von Neumann (LvN) equation(47, 45)... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

Note that the Liouville matrix, iL + R + K may not be Hermitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If we allow complex numbers in the equation, (12) is a general result. Since A is a diagonal matrix we can expand in terms of the individual eigenvalues, Xj. We can also apply U ( backwards )... 

In the above equation, q has been considered to have only one component that is along the -direction. L is the Hermitian Liouville operator, Q = 1 — P, where P is the projector onto the dynamical quantity A . Aa is component of a four-vector of particle current density and particle density, defined as... 

The matrices in equation (35) for a system of n spins of 1/2 have dimensions of 22n. This means that, for example, a four-spin system must be considered within a space of 256 dimensions. If we deal with the motion of a spin system in a static magnetic field (as in pulse-type experiments), significant simplifications are possible owing to the rules of commutation. Namely, if the Hermitian operators A and 6 commute in Hilbert space, then all the corresponding superoperators AL, AR, AD, BL, BR, and BD in Liouville space also commute. The proof of this is given in reference (12). In Hilbert space, the following commutation takes place ... 

Equation (116) has a form which is similar to that of the equation (35) of motion for non-exchanging spin systems. The analogy is even closer, as is shown later, since a judicious renormalization of the vectors in the composite Liouville space can convert equation (116) into one in which all the superoperators become Hermitian. Firstly we wish to draw attention to some of the properties of the exchange superoperator X. ... 

In the renormalized composite Liouville space the superoperator F , defined by equation (121), commutes with all the (Hermitian) superoperators Hf, R, and X. Therefore, upon the proper rearrangement of the basis set in this space, one can obtain a factorization of the equation (137) of motion into blocks which are connected with individual eigenvalues of the superoperator F°. This resembles the analogous procedure in the case of static NMR spectra, i.e. those for non-exchanging spin systems (Section II.E.2). The equations for the free induction decay M ID and for the lineshape of an unsaturated steady-state spectrum, in terms of quantities from composite Liouville space, are therefore obtained for exchanging spin systems in a way which is analogous to that for non-exchanging systems (Section II.F). 

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form... 

According to this requirement, we could give the actual form of the equation of motion for the density operator in the Schrodinger picture. We will see that this equation corresponds to the Liouville-von Neumann equation in the case of dissipative processes. From Equations (113) and (119), it follows that the density operator in the Schrodinger picture could be written by a Hermitian operator in the form... 

A simple but nonrigorous version of this proof is the following. Since the Liouville operator is Hermitian its eigenvalues A are real and its eigenfunctions x(r) are orthogonal where... 

The main reason for introducing this scalar product is that the Liouville operator L defined by eq 2.19 is Hermitian with respect to this scalar product ... 

Any operator on functions of F transforms one vector in Liouville space into another. Of particular importance to us are the operators L and It is easy to show by an integration by parts that L is a linear Hermitian operator in Liouville space, that is,... 

Since (A Af ) = (Ay, Af), the matrix x is a Hermitian matrix, so that x = X-The properties Ai,..., A define a subspace of Liouville space. This subspace is the set of all vectors that can be expressed as linear combinations of Aj,..., Am. Let us now determine the projection operator P that projects a vector onto this M-dimensional subspace. The projection operator must satisfy the conditions PA = A and P = P. Note that... 

Eq (1.6) serves to define the Liouville operator L the factor i = /-I is introduced to make this operator be Hermitian The formal solution to (1.6) is... 

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