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# notes06 - 5.73 Lecture#6 6-1 Last time Normalization of...

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6 - 1 5.73 Lecture #6 updated 9/12/02 11:39 AM WKB Quantization Condition x _ (E) x + p( x)d x = h 2 n + 12 () n = 0,1, 1. identities 2. trick using box normalization often needed - alternate method via JWKB next lecture box # states # particles x ψψψψ δδδ δθ δ kpE L L dn dE ,,, . . / 3 4 1 1. V(x) = α x linear potential solve in momentum representation, φ (p), and take F.T. to ψ (x) Airy functions 2. Semi-classical (JWKB) approx. for ψ (x) for box normalization ] Next lecture *( ) ( ( ) ) * ( ) ( ) exp ( ) / / px E Vx m xp x i px dx c x =− [] 2 ψ h envelope variable * visualize y(x) as plane wave with x-dependent wave vector * useful for evaluating stationary phase integrals (localization, causality) **** splicing across classical (E > V)|| forbidden (E < V) Last time: Normalization of eigenfunctions which belong to continuously variable eigenvalues.

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6 - 2 5.73 Lecture #6 updated 9/12/02 11:39 AM Solve in momentum representation (a sometimes useful trick) Schr. Eq. d φ (p) dp =− i h α E p 2 2m () φ solution φ = Ne ap + bp 3 gives constant times φ (p ) gives p 2 times φ plug into Schr. Eq. and identify, term-by-term, to get a = iE h α b = i 6 h m φ = Nexp i h α Ep p 3 /6m easy? √√ H p m x x xi p i H m d dx xH p m i d dp =+ →→ [] = + = + 2 22 2 2 2 α αα coordinate representation momentum representation xx pp p i note x, p in both representations - prove this? 2nd order 1st order - easier! h h h h h Linear Potential. V(x) = α x φφ *( ) ( ) ! = 1 ∴= ! N1
6 - 3 5.73 Lecture #6 updated 9/12/02 11:39 AM If we insist on working in the ψ (x) picture, we must perform a Fourier Transform ψφ ψ α α () ( ) exp / / xN e p d p xN i pxE p md p ipx =′ + −∞ −∞ h h 12 44 43 444 3 6 odd function of p: O(p) ψ (x) = N −∞ cos ( α x E)p + p 3 /6m h α dp Surprise! This is a named (Airy) and tabulated integral * numerical tables for x near turning point i.e., x E/ α analytic asymptotic” functions for x far from turning point useful for deriving f(q.n.) and for matching across boundaries. Now p is an observable, so it must be real. Thus φ (p) is defined for all (real) p and is oscillatory in p for all p. NEVER exponentially increasing or decreasing! IT IS STRANGE THAT φ (p) does not distinguish between classically allowed and forbidden regions. IS THIS REALLY STRANGE? If we allow p to be imaginary in order to deal with classically forbidden regions, φ (p) becomes an increasing or decreasing exponential. ei O p d p i even odd θ θθ =+ = −∞ cos sin sin ( ) { { 0 Ai z s sz ds ( ) cos / + 3 0 3 * zeroes of Airy functions [Ai(z i )=0] and of derivatives of Airy functions [Ai (z i )=0] are tabulated. (Useful for matching across center of potentials with definite even or odd symmetry.) [Two kinds of Airy functions, Ai and Bi.] since O(p) is odd wrt p –p.

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6 - 4 5.73 Lecture #6 updated 9/12/02 11:39 AM Ai(z) −1 / 2 0 cos s 3 3 + sz
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## This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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notes06 - 5.73 Lecture#6 6-1 Last time Normalization of...

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