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Read e-book online Advances in quantum chemistry. / Volume 4 PDF

By Per-Olov Lowdin

ISBN-10: 0120348047

ISBN-13: 9780120348046

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Extra info for Advances in quantum chemistry. / Volume 4

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42). 42) must hold for every f in D’, not only for f =fo . Thus we have arrived at one of the self-adjoint extensions of Do which we encountered earlier. ,[- co, co] which we denote for brevity by Q2,. This problem is much simplified by the fact that no boundary conditions are necessary, because for all functions which satisfy condition (A) in Q Z m , lim,,,,f(x) = lim,= - m f ( x ) = 0. 49) for all pairs of functionsf and g which satisfy condition (A). Therefore D is defined by the single requirement of condition (A), moreover Dt = D, the operator is self-adjoint.

Lengyel 30 or the spectral theorem. 4) has nontrivial solutions for certain specific values of the variable A. These values Al, A,, ... are called the eigenvalues of the matrix. 4) are called eigenvectors. An eigenvector X’ = {xii, x 2 j , ... ,x i } satisfies the equations n C k= 1 = Ijxi ( i = 1,2, ... 5) which is written in the compact form HX’ = A j X j . The solutions of the Eq. 3) contain a complete orthonormal system of vectors. When these vectors are introduced to define the axes of a coordinate system in the n-dimensioned space, the Hermitean matrix assumes the following diagonal form : 1 ; j.

When the domain of an operator is enlarged, the domain of its adjoint is 46 Bela A. Lengyel either decreased or it remains the same. In symbols: A 3 B implies Bt 3 A t . The question immediately arises, what symmetric operators can be extended so that they become self-adjoint? In order to discuss this question and related other questions we have to digress and introduce the concept of closure. The next section is devoted to this topic. Exercises 1. n. set {q,}in &,define a linear operator H formally by the equations H q , = n q , ,n = 1,2,3, ...

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