08general - P460 - math concepts 1 General Structure of...

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Unformatted text preview: P460 - math concepts 1 General Structure of Wave Mechanics (Ch. 5) Sections 5-1 to 5-3 review items covered previously use Hermitian operators to represent observables (H,p,x) eigenvalues of Hermitian operators are real and give the expectation values eigenvectors for different eigenvalues are orthogonal and form a complete set of states any function in the space can be formed from a linear series of the eigenfunctions some variables are conjugate (position, momentum) and one can transform from one to the other and solve the problem in eithers space x d x i dx p p e x u C t x product dot dx u u u u j i x u ions eigenfunct op iEt N N ij j i j i i ) ( ) ( ) , ( | | ) ( : * * / * = >= < = = >= >=< < h h P460 - math concepts 2 Notation there is a very compact format (Dirac notation) that is often used |i> = |u i > = eigenfunction < |> is a dot product between 2 function |i><j| is an outer product (a matrix). For example a rotation between two different basis if an index is repeated there is an implied sum operator projection n n n n n C x u C x product dot dx u u u u j i x u ions eigenfunct N N N ij j i j i i = = = = >= >=< < ) ( ) ( | | ) ( : * P460 - math concepts 3 Degeneracy (Ch. 5-4) If two different eigenfunctions have the same eigenvalue they...
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This note was uploaded on 10/02/2009 for the course PHYS 460 taught by Professor Johnson,c during the Spring '08 term at Northern Illinois University.

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08general - P460 - math concepts 1 General Structure of...

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