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08operators - Operator methods in Quantum Mechanics Section...

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P460 - operators and H.O. 1 Operator methods in Quantum Mechanics Section 6-1 outlines some formalism – don’t get lost; much you understand define ket and bra vectors and dot product add in operators to this formalism. Let A be an operator dx x x x x x x ) ( ) ( | | ) ( | ) ( * * ψ φ ψ φ φ φ φ ψ ψ ψ = dx A dx A A A dx A A A A A dag dag ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ ψ = = = = = = * * * ) ( | |
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P460 - operators and H.O. 2 Orthonormal States can usually define a set of orthonormal states |n> (eigenfunctions). Can rotate to this basis (diagonalize operater) any other function can be made from these identical to 2D vectors . 1 oper unit n n n n n C n C n m states l orthonorma n n n n n n mn = = = = = = ψ ψ ψ ψ δ ( ) ( ) ( ) = + = = = + = + = 1 0 0 1 1 0 1 0 0 1 0 1 1 2 0 1 3 2 ˆ 1 0 3 0 1 2 3 2 ˆ 3 ˆ 2 n n i r j i r n r r
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P460 - operators and H.O. 3 Projection operator this defines the projection operator P n which when it acts on an arbitrary state projects it into the state |n> so “projection” along “vector” n. Again for 2D vectors ) ( 1 1 ) ( matrix P n n as P n n m m P P matrix diagnonal n n P n n n n n n mn n m n n = = = = = = δ
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