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

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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 either’s “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 ψ ψ ψ ψ ψ δ ) ( ) ( ) , ( | | ) ( : * * / * ∂ ∂ − = >= < = ⇒ = >= >=< < ∫ ∫ ∑ ∫ − h h P460 - math concepts 2 Notation • there is a very compact format (Dirac notation) that is often used • |i> = |u i > = eigenfunction • < χ|φ> 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 ≡ = = ⇒ = ⇒ = >= >=< < ∑ ∫ ψ ψ ψ δ ) ( ) ( | | ) ( : * P460 - math concepts 3 Degeneracy (Ch. 5-4) • If two different eigenfunctions have the same eigenvalue they...
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