Chapter3_2 - PHY4604 R D Field Obsevables and Vector Spaces...

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PHY4604 R. D. Field Department of Physics Chapter3_2.doc University of Florida Obsevables and Vector Spaces Eigenvalue Equation: Consider the case of an hermitian operator A op acting on a set of wave functions, | ψ n >, such that A op | ψ n > = a n | ψ n > where a n is a constant. Note a n is real and < ψ i | ψ j > = δ ij ( i.e. orthonormal set). Completeness: If the ensemble of eigenkets | ψ n > spans the entire space ( i.e. any “ket” of finite norm can be expanded in a series of these “kets”), then they are said to form a complete set and the hermitian operator A op is called an observable . Vector Space (ordinary vectors): In three-dimensional space any vector can be written as a superposition of the three unit vectors, z a y a x a a ˆ ˆ ˆ 3 2 1 + + = r , where a 1 , a 2 , a 3 are real numbers. The three unit vectors form an orthonormal bases. The simplest way to represent a vector is to specify the three constants a i with respect to the basis, = 3 2 1 a a a a r and
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