ALTERNATIVE REPRESENTATIONS OF QUANTUM MECHANICS

The derivations made so far were done in terms of the wave functions (vectors). In fact all the basic tools of the quantum theory used throughout this book are covered by this brief account. However, it may be practical to have a variety of representations for the same set of basic techniques in different incarnations. Not adding too much either to pragmatic numerical tools or to a deeper understanding of what is going on, these tools are useful for getting general relations which are an important part of the present book and for a more economical representation of the variables of the problems considered here. For that reason we review them below. 

Taking a vector (function) f ) in the space of the allowable wave functions (e.g. integrable with its square and normalized to unity) allows us to construct an operator Pv which acts on an arbitrary vector D) (function) as  

12 Presentation in this section follows the route presented in the brilliant lectures delivered by one of the author s teachers Dr. V.I. Pupyshev of the Chemistry Department of the Moscow State University during last 30 years, but which appeared in a printed form [V.I. Pupyshev, Additional chapters of molecular quantum mechanics, Parts 1-3. Moscow University Publishers [in Russian], 2008] during the time of proofreading of the present book. 

If an orthonormalized basis is given, the vector p) is defined by its expansion over it  

It projects to the subspace spanned by p) and ip). This construct is extended to any number of orthonormal vectors. 

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Approximate techniques for alternative representations of quantum mechanics... 

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