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Essential Quantum Mechanics$
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Gary Bowman

Print publication date: 2007

Print ISBN-13: 9780199228928

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780199228928.001.0001

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The Structure of Quantum States

The Structure of Quantum States

(p.36) 4 The Structure of Quantum States
Essential Quantum Mechanics

Gary E. Bowman

Oxford University Press

The quantum state is the fundamental entity in quantum mechanics. This chapter develops the basic mathematical structure of quantum states by developing the correspondence between states and ordinary vectors. The familiar scalar or dot product of vectors is used to develop the concepts of projection, orthonormality, linear independence, and expansion over a complete set of basis vectors. These ideas are then extended from vectors in a vector space to functions in a function space. Dirac's bra-ket notation is introduced, and the concepts developed for ordinary vectors are revisited in the context of quantum states and Dirac notation. The central quantum concept of a representation is then explored, and the relation between representations and superpositions is illustrated with the infinite potential well. The final topic is representational freedom: the fact that quantum mechanics remains valid regardless of the representation chosen for calculation.

Keywords:   vector, scalar product, orthonormal, Dirac notation, representation, superposition, infinite potential well

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