Chapter3_3 - dual space Projection Operators If | α> is a normalized “ket” vector Then the operator(P i op = | ψ i>< ψ i | projects

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PHY4604 R. D. Field Department of Physics Chapter3_3.doc University of Florida Projection Operators and Completeness Eigenvalue Equation: Consider 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). Expansion Theorem: Any “ket” ( i.e. wave function) with finite norm can be written as a superposition of “eigenkets” of the observable A op which form the basis ( i.e. representation), > + + > + > >= n n a a a ψ α | | | | 2 2 1 1 L , where the constants a i are complex numbers and where n may go to infinity. The simplest way to represent the “ket” vectors is to specify the constants a k with respect to the basis as a column vector, >= n a a a M 2 1 | . The corresponding “bra” is a row vector ) ( ) (| | 2 1 = > = < n a a a a L . The collection of “bra” vectors forms another vector space (the so-called
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Unformatted text preview: dual space ). Projection Operators: If | α > is a ( normalized ) “ket” vector. Then the operator (P i ) op = | ψ i >< ψ i | projects out the portion of any “ket” vector that “lies along” | ψ i > . Namely, ( P i ) op | α > = | ψ i >< ψ i | α > = a i | ψ i > , where a i = < ψ i | α > . The operator ( P i ) op is the projection operator onto the one-dimensional-space spanned by | ψ i > . The sum over all the projection operators ( P i ) op gives 1 | | ) ( 1 1 = >< = ∑ ∑ = = n i i i n i i P where 1 is the identity operator. This is the condition necessary for | ψ i > to form a complete set of states. It is true since > = > >< ∑ = | | | 1 n i i i . Completeness condition!...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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