notes11 - 5.73 Lecture #11 11 - 1 Eigenvalues,...

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11 - 1 5.73 Lecture #11 updated 10/1/02 11:00 AM should have read CDTL pages 94-144 Last time: bra a a a a NN kk N N k 1 1 1 1 1 0 0 ** # () × = = φ φ ket matrix complex M O φ i AB φ j i A φ k φ k 1 12 43 4 B φ j k = A ik B kj k = AB ij at end of lecture Eigenvalues, Eigenvectors, and Discrete Variable Representation (DVR) ψφ ψ φψ ψ φ φφ in basis set i {} = = = 0 1 0 1 2 M M M M a a a a N jj i 1
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11 - 2 5.73 Lecture #11 updated 10/1/02 11:00 AM Today: eigenvalues of a matrix – what are they? how do we get them? (secular equation). Why do we need them? eigenvectors – how do we get them? Arbitrary V(x) in Harmonic Oscillator Basis Set (DVR) What is the connection between the Schrödinger and Heisenberg representations? ψψ δ ii xx x () , = = 00 eigenfunction of x with eigenvalue x 0 Using this formulation for ψ i (x), you can go freely (and rigorously) between the Schrödinger and Heisenberg approaches. 1 == k kk d x
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11 - 3 5.73 Lecture #11 updated 10/1/02 11:00 AM Schr. Eq. is an eigenvalue equation  ψ = a ψ in matrix language A ψψ ψ ψ ψ ii i N aA a a a == = 1 2 1 0 0 1 0 0 O M but that is the eigen-basis representation – a special representation! What about an arbitrary representation? Call it the φ representation. *** *** A A ci a i i N i i i φφ φ = ∑∑ = 1 as transformation on each Eigenvalue equation N unknown coefficients c to N How to determine c and a ? i i {} = {} i1 Secular Eqn. derive it. first, left multiply by j φ A A one equation ji ji φ φ δ δ cac j iac ca i i i i j i j i N =− [] = 0 1 N unknowns next, multiply original equation by k φ . 0 1 = c A a nother ik i i k i N φ φ δ a equation etc. for all N linear homogeneous equations in N unknowns – Condition that a nontrivial (i.e. not all 0’s) solution exists is that determinant of coefficients = 0.
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notes11 - 5.73 Lecture #11 11 - 1 Eigenvalues,...

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