Basic mathematical constructions

A physical system, which may be part of a larger system, is associated with a linear vector space whose elements are ket vectors 

For each space there is a dual space of bra vectors 

The letters A and B denote quantities characterising the vectors. They refer to dynamical aspects of the state of the system. The vectors are called state vectors, a term which we abbreviate to states. The length of a state vector, considered apart from related state vectors, has no physical significance. We are free to choose it, a process known as normalisation. 

The states are of two kinds, which require rather different treatments. The specifying quantities A may be members of a discrete set counted by an integer /, in which case the state is denoted 

The spaces of interest in physics are spanned by the eigenstates a ) of real (that is self-adjoint) linear operators a, whose eigenvalues a are real. Such operators are called observables. The eigenvalue equation is 

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The basic mathematical tool used in the construction of approximate eigenfunctions is the variation theorem. This theorem asserts that for any square integrable function O which is of a definite and allowed symmetry for the Hamiltonian, the quotient... 

El theory In all materials (plastics, metals, wood, etc.) elementary mechanical theory demonstrates that some shapes resist deformation from external loads. This phenomenon stems from the basic physical fact that deformation in beam or sheet sections depends upon the mathematical product of the modulus of elasticity (E) and the moment of inertia (I), commonly expressed as EL This theory has been applied to many different constructions including sandwich panels. 

Although the above discussion assumes that all MOs are occupied by two electrons, it turns out that the basic ideas can be extended to open-shell molecules in which there are unequal numbers of electrons in the two spin states. Without showing the complicated mathematics, we will show how the wavefunction can be determined by constructing two Fock matrices for each spin state and then solving two sets of coupled Roothaan equations ... 

A variant of the method discussed in this chapter has been proposed by C. A. R. Hoare using a set of axioms and rules of inference to establish partial correctness of programs. The method of Hoare appears more flexible in that axioms and rules can be introduced to cover various constructs of particular programming languages and their implementations, but also appears, at least to this author, even more cumbersome and unwieldy than the Floyd-Manna-King approach when applied to simple flowchart-like programs. The formal mathematical justification for both approaches is the same. Basically, the approach used to date employs "forward substitution" from hypothesis assertion to conclusion assertion while the Hoare... 

The Eulerian continuum approach is basically an extension of the mathematical formulation of the fluid dynamics for a single phase to a multiphase. However, since neither the fluid phase nor the particle phase is actually continuous throughout the system at any moment, ways to construct a continuum of each phase have to be established. The transport properties of each pseudocontinuous phase, or the turbulence models of each phase in the case of turbulent gas-solid flows, need to be determined. In addition, the phase interactions must be expressed in continuous forms. 

The present chapter is not meant to be exhaustive. Rather, an attempt has been made to introduce the reader to the major concepts and tools used by catalytic reaction engineers. Section 2 gives a review of the most important reactor types. This is deliberately not done in a narrative way, i.e. by describing the physical appearance of chemical reactors. Emphasis is placed on the way mathematical model equations are constructed for each category of reactor. Basically, this boils down to the application of the conservation laws of mass, energy and possibly momentum. Section 7.3 presents an analysis of the effect of the finite rate at which reaction components and/or heat are supplied to or removed from the locus of reaction, i.e. the catalytic site. Finally, the material developed in Sections 7.2 and 7.3 is applied to the design of laboratory reactors and to the analysis of rate data in Section 7.4. 

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