Wigner’s formulation

The final and most fundamental impact of Wigner s paper lies in the impetus that it gave to formulating a rigorous quantum theory of the chemical rate constant, without the need to solve the full state to state scattering problem. Early work by McLafferty and Pechukas concluded that can be no... 

In principle, the correspondence between the two theories is not complete, because scattering theory is the more general formulation. For our purposes, however, the fact that the applications to atomic physics obtained by both methods are quite consistent with each other is an important and useful conclusion. The same result and connections have been obtained independently by Komninos and Nicolaides [378]. Both [373] and [378] noted that the derivation of MQDT from Wigner s scattering theory establishes its basic structure and theorems without special assumptions about the asymptotic forms of wavefunctions. The approach of Komninos and Nicolaides [378] is designed for applications involving Hartree-Fock and multiconfigurational Hartree-Fock bases. In the present exposition, we follow the approach and notation of Lane [379] and others [380, 381], who exploit the analytic K-matrix formalism and include photon widths explicitly when interferences occur. 

Advancements in TST have been well documented in the literature over the past 23 years [9-16]. Much of the work on TST has focused on understanding the dynamical foundations of the theory and the extension of the theory to allow for quantitatively accurate estimates of rate constants. Advancements in these areas can be attributed to the fact that the TST expression for the classical equilibrium rate constant can be formulated by making a single approximation, Wigner s fundamental assumption. 

TST has also been widely used to treat reactions in condensed phases. Wigner s dynamical perspective has particularly had an impact on the extension of TST to reactions in liquids. Most applications to liquid-phase reactions have used the thermodynamic formulation of TST [60], which includes the effects of the condensed phase on reaction free energies in an approximate manner. Chandler [61] provided a more rigorous formulation of classical TST for liquids. The new element introduced by the liquid phase is collisions of solvent molecules with the reacting species that can lead to recrossings of the dividing surface and a breakdown of the fundamental assumption. A recent review [16] documents many more advances in the extension of TST to the kinetics of condensed-phase processes. 

The function Dxp is composed of Dira functions D = 2b gb 5b where 2b gb = 1 gb => 0 and the 5s are forbidden by Planck s law of the finiteness of the quantum of action (which may be formulated as the indeterminacy principle). This may be corrected for by the so-called Wigner transformation which transforms the expectation value linear form to U = / Hx x DXx where the coordinates x occur twice, independently, so that H and D become matrices. Since the Wigner transformation must lead to indeterminacy, it is closely related to a Fourier transformation and both matrices H and D become hermitian (Bopp, 1961). The dyads of hermitian matrices may be written j/x i//x and we see that their contribution u to the expected energy becomes u = / jj Hi//, therefore if we choose i//x as eigenvectors of H we see that we have in fact only discrete possible states in agreement with indeterminacy. 

The analytic form of the pump and dump pulses in the optimal phase-unlocked pump-dump control is />( >)( )= p( >)(f) exp (—/cOe f)- -exp megt), where Tipp) is a slowly varying envelope of the helds, and (s>eg is the energy difference between the minima of the excited and the ground states. The objective in the ground state is represented in the Wigner formulation by an operator A = A(r) g)(g. A(r) is the Wigner transform of the objective in the phase space T = of coordinates and momenta, and... 

An important theorem which derives from Schur s lemma is the fundamental theorem for irreducible representations (3). This can be formulated as the Wigner-Eckart theorem, Eq. (3). In order to obtain this theorem in the present formulation, each irreducible representation must be chosen in identically the same form each time it occurs, rather than in an equivalent form. Therefore the irreducible representations are conveniently generated in standard form by applying the operators of the symmetry group to a properly cho.sen set of standard bases for the irreducible representations. 

During the three productive years of a postdoctoral stay in Mark s Laboratory, Robert extended Einstein s equation (originally derived for linear stress gradient) to parabolic Poiseuille flow. There were excursions with Eirich into kinetic theory and viscosity of gaseous paraffins, as well as viscosity, surface tension, and heat of vaporization correlations of chain molecular fluids. The latter made use of the recently formulated transition-state theory of Eyring, Polanyi, and Wigner. 

The assumptions for the Wigner condition do not seem to be valid at the present moment. Jastrow interprets the high energy scattering data in terms of a repulsive core. Feshbach s boundary condition formulation of the two-body data implies velocity dependent forces (in that the boundary conditions are energy independent). Feshbach s solutions are also not in agreement with the Serber mixture ". 

This paper is widely regarded as the origin of Slater s theory of unimolecular reactions, it seems to me ironic that Polanyi Wigner were so close to a successful formulation of unimolecular reactions, but lacking the necessary theoretical apparatus at that time, they were forced to resort to a mechanical analogy one stage too soon the end result, despite valiant efforts by Noel Slater, is a morass. 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodynamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic description. The transition from quantum to cla.s.sic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

Let us focus on the origin of the principal idea of the J-T effect. Before its final formulation by Jahn and Teller, first the Teller s student Renner [33] was inspired with the von Neumann-Wigner theorem about crossing electronic terms [34] Electronic states of a diatomic molecule do not cross, unless permitted by symmetry . Only if the states have different symmetry, they can cross. 

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