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Ideas of Quantum Chemistry P16

Ideas of Quantum Chemistry P16 shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field. | 116 3. Beyond the Schrödinger Equation spin concept will be shown later on 1 1 2 01 2Z where the first two components i and i r2 functions of class Q for reasons that will become clear in a moment are called large components are hidden in vector ty while the two small components 0i and 2 functions of class Q 21 are labelled by vector . Vectors ty and are called the spinors. How to operate the N-component spinor for N 4 we have called them bispinors Let us construct the proper Hilbert space for the N-component spinors. As usual p. 895 first we will define the sum of two spinors in the following way 1 1 S N 1 1 2 2 Wtf N and then the product of the spinor by a number y 1 y t y T 7 2 N WaJ Next we check that the spinors form an Abelian group with respect to the above defined addition cf. Appendix C p. 903 and that the conditions for the vector space are fulfilled Appendix B . Then we define the scalar product of two spinors N W i i i 1 where the scalar products j j are defined as usual in the Hilbert space of class Q functions. Then using the scalar product T we define the distance between two spinors - TJI - T d - T and afterwards the concept of the Cauchy series the distances between the consecutive terms tend to zero . The Hilbert space of spinors will contain all the linear combinations of the spinors together with the limits of all the convergent Cauchy series. 21 It will be shown that in the non-relativistic approximation the large components reduce to the wave function known from the Schrodinger equation and the small components vanish. In eq. 3.54 the constant E as well as the function V individually multiply each component of the bispinor while a n axnx ay y aznz denotes the dot product of the matrices a r x y z by the operators y in the absence of the electromagnetic field it is simply the momentum operator component see p. 962 . The matrix p is multiplied by the constant moc2 then by the bispinor 3.3 The Dirac equation 117 An operator acting on a spinor .