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385appA

# 385appA - PHYS 385 Appendix A Eigenvalues and...

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PHYS 385 Appendix A - Eigenvalues and eigenfunctions A - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Appendix A - Eigenvalues and eigenfunctions What's important : eigenvalues and eigenfunctions Hermitian operators Text : Gasiorowicz, Chap. ?? Some Eigenvalue theorems In our discussion of the hydrogen atom, we made several statements about orthogonality of eigenfunctions and whether there could be simultaneous eigenvalues in specific cases. We now prove these results in general. 1. Eigenfunctions belonging to different eigenvalues of a Hermitian operator are orthogonal. Proof: F hermitian operator with eigenfunctions ψ 2 and ψ j so that F ψ i = f i ψ i ; F ψ j = f j ψ j where f eigenvalues. Then < ψ j F ψ i > = f i < ψ j ψ i > But < ψ j F ψ i > = < ψ i F ψ j >* = f j * < ψ i ψ j >* = f j * < ψ j ψ i > Hence, f i < ψ j ψ i > = f j * < ψ j ψ i > Since F is hermitian, then f’s are real. (f i – f j ) < ψ j ψ i > = 0

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