Schwinger’s principle

Chapter 8 gives a full account of Schwinger s principle of stationary action and of how, through its generalization, one obtains a definition of a quantum subsystem and a description of its properties. 

We are now in a position to perform a generalized variation of the action integral for an open system to demonstrate that Schwinger s principle of stationary action can be extended in such a manner as to provide a quantum definition of an atom in a molecule. We shall be considering the change in the atomic action integral 2] of eqn (8.111) ensuing from variations... 

Bader has shown that the topological partitioning of the molecules into atomic basins coincides with the requirements of formulating quantum mechanics for open systems [93], and in this way all the so-called theorems of quantum mechanics can be derived for an open system [94], Furthermore, the zero-flux condition, Eq. 1, turns out to be the necessary constraint for the application of Schwinger s principle of stationary action [95] to a part of a quantum system [93], The successful application of QTAIM to numerous chemical problems has thus deep physical roots since it is a theory which expands and generalises quantum mechanics themselves to include open and total systems, both treated on equal formal footing. 

Properties of a proper open system 12 are defined by Heisenberg s equation of motion obtained from the variation of the state vector within the system and on its boundaries [1,4], in the manner determined by Schwinger s principle of stationary action [5]. For a stationary state, the equation of motion for an observable G is given in Equation (2),... 

We are interested in molecules in stationary states and in this case Schwinger s principle takes on a particularly simple form. In this instance, it yields Schrodinger s equation for a stationary state, equation 11, and equation 14 for the variation in Schrodinger s energy functional G[T ],... 

It is through a generalization of Schwinger s principle that one obtains a prediction of the properties of an atom in a molecule. The generalization is possible only if the atom is defined to be a region of space bounded by a surface which satisfies the zero flux boundary condition, a condition repeated here as equation 15,... 

These rules are a part of the formalism of quantum mechanics that is recovered from Schwinger s action principle. 

The analoguous behiaviour of the Poisson bracket and the commutator has been used to establishcorrespondence between classical and quantum mechanics. It is, however/shown in the next section following the derivation of Schwinger s quantum action principle that the correspondence goes deeper and that the analogous behaviour of the Poisson bracket and commutator is a consequence of the properties of infinitesimal canonical transformations which are common to both mechanics. 

Schwinger s quantum action principle (1951) is developed in Section 8.2, and a statement of this principle, in which the action integral and Lagrange... 

The precise connection with finite dimensional matrix formulas obtains simply from Lowdin s inner and outer projections [21, 22], see more below, or equivalently from the corresponding Hylleraas-Lippmann-Schwinger-type variational principles [24, 25]. For instance, if we restrict our operator representations to an n-dimensional linear manifold (orthonormal for simplicity) defined by... 

The three variational principles in common use in scattering theory are due to Kohn [9], Schwinger [11] and Newton [12]. Two of these variational principles, those due to Kohn and Newton, have been successfully developed and applied to reactive scattering problems in recent years there is the S-matrix Kohn method of Zhang, Chu, and Miller, the related log derivative Kohn method of Manolopoulos, D Mello, and Wyatt and the L - Amplitude Density Generalized Newton Variational Principle (L -AD GNVP) method of Schwenke, Kouri, and Truhlar. 

The Kohn variational principle is perhaps the simplest of the three scattering variational principles mentioned above [9]. In particular, it requires that one calculate matrix elements only over the total Hamiltonian H of the system, and not over the Green s function Gq E) of some reference Hamiltonian Hq. While matrix elements of H between energy-independent basis functions are also energy-independent, all matrix elements of G E) have to be re-evaluated at each new scattering energy E. The Kohn variational principle is therefore somewhat easier to apply than the Schwinger and Newton... 

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