Hilbert space introduction

Halmos, P.R. (1957) Introduction to Hilbert Space (and the Theory of Spectral Multiplicity) (2nd edn.), Chelsea, New York. 

As stated in the introduction, we present the derivation of an extended BO approximate equation for a Hilbert space of arbitary dimensions, for a situation where all the surfaces including the ground-state surface, have a degeneracy along a single line (e.g., a conical intersection) with the excited states. In a two-state problem, this kind of derivation can be done with an arbitary t matrix. On the contrary, such derivation for an N > 2 dimensional case has been performed with some limits to the elements of the r matrix. Hence, in this sence the present derivation is not general but hoped that with some additional assumptions it will be applicable for more general cases. 

A manner to do away with the problem is to introduce appropriate algorithms in the sense that mappings from real space to Hilbert space can be defined. The generalized electronic diabatic, GED approach fulfils this constraint while the BO scheme as given by Meyer [2] does not due to an early introduction of center-of-mass coordinates and rotating frame. The standard BO takes a typical molecule as an object description. Similarly, the wave function is taken to describe the electrons and nuclei. Thus, the adiabatic picture follows. The electrons instantaneously follow the position of the nuclei. This picture requires the system to be always in the ground state. 

DeM] Debnatb, L. and P. Mikusinski, Introduction to Hilbert Spaces with Applications, Second Edition, Academic Press, San Diego, 1999. 

Murray, F., [1] An introduction to linear transformations in Hilbert space, Princeton, 1941. Nagumo, M., [1] Einige analytische Untersuchungen m linearen metrischen Rmgen, Jap. J. Math. 13 (1936), 61-80. 

We will discuss the Floquet approach from two different points of view. In the first one, discussed in Section II.A, the Floquet formalism is just a mathematically convenient tool that allows us to transform the Schrodinger equation with a time-dependent Hamiltonian into an equivalent equation with a time-independent Hamiltonian. This new equation is defined on an enlarged Hilbert space. The time dependence has been substituted by the introduction of one auxiliary dynamical variable for each laser frequency. The second point of... 

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]. 

As discussed subsequently, introduction of the standard p, q representation in classical mechanics, and of the Wigner-Weyl representation in quantum mechanics, defines densities p(p,q) = (p,q p) that both lie in the same Hilbert space. Thus, the essential difference between quantum and classical mechanics... 

The classical Liouville equation does have an equivalent in quantum mechanics, which is needed for a consistent description of quantum statistical mechanics the quantum Liouville equation. Equilibrium quantum statistical mechanics requires the introduction of the density operator on an appropriate Hilbert space, and the quantum liouvUle equation for the density operator is a logical and necessary extension of the Schrodinger equation. The quantum Liouville equation can even be written, formally at least, in a form that resembles its classical counterpart. It allows for some weak and almost internally consistent form of dissipative dynamics, known as the Redfield theory, which finds its main use in relating NMR relaxation times to spectral densities arising from solvent fluctuations, although in recent... 

I now consider statement 3 How should an extension of dynamics be understood In the MPC theory the problem does not exist For the intrinsically stochastic systems there is no need for modifying the laws of dynamics. As for the LPS theory, one notes the presence of two essentially new concepts. The introduction of non-Hilbert functional spaces only concerns the definition of the states of the dynamical system, and not at all the law governing their evolution. It is an important precision introduced in statistical mechanics. The extension of dynamics thus only appears in the operation of regularization of the resonances. This step is also the one that is most difficult to justify rigorously it is related to the (practical) necessity to use perturbation calculus (see Appendix). 

For an introduction to Laguerre polynomials, see pages 93-97 of Courant and Hilbert (1956), referenced in Chapter 4. The functions Ajirj, r]") may be regarded as from the tensor product space of L2[0, oo with itself, having an inner product as defined in (6.2.11). For a definition of the tensor product of function spaces, see Ramkrishna and Amundson (1985), referenced in Chapter 4. 

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