notes07 - 5.73 Lecture #7 7-1 JWKB QUANTIZATION CONDITION...

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5.73 Lecture #7 7 - 1 JWKB QUANTIZATION CONDITION Last time : 1. Vx ( ) = α x φ (p) = Nexp h i α ( Ep p 3 6m ) x ψ () = Ai(z) * zeroes of Ai, Ai * tables of Ai (and Bi) * asymptotic forms far from turning points 2. Semi-Classical Approximation for ψ (x) * p(x) = [ ( E V(x) ) 2m ] 1/2 modifies classical to make it QM wavefunction “classical * ψ (x) = p(x) 1/2 wavefunction” x exp ± i p(x)dx ′ h 4 c envelope± 1243 adjustable phase for wiggly-variable k(x) boundary conditions ψ without differential equation qualitative behavior of integrals (stationary phase) d λ *± validity: << 1 —valid not too near turning point. dx [One reason for using semi-classical wavefunctions is that we often need to evaluate integrals * ˆ ˆ O of the type ψ ip ψ j dx. If O p is a slow function of x, the phase factor is i d   exp h [ px ) ) ] dx . Take dx = 0 to find x s p . δ x is range about x s p over which j ( i ( .. 2 ( ) δ x.] phase changes by ±π /. Integral is equal to Ix sp Logical Structure of pages 6-11 to 6-14 (not covered in lecture): 1.± ψ JWKB not valid (it blows up) near turning point — can’t match ψ ’s on either side of turning point. 2. Near a turning point, x ² (E), every well-behaved V(x) looks linear V(x) + (E) ) + dV ( dx x = x + ( x x + ) first term in a Taylor series. This makes it possible to use Airy functions for any V(x) near turning point. updated 9/16/02 11:29 AM
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5.73 Lecture #7 7 - 2 3. asymptotic-Airy functions have matched amplitudes (and phase) across validity gap straddling the turning point. 4. ψ JWKB for a linear V(x) is identical to asymptotic-Airy! TODAY 1. Summary of regions of validity for Airy, a-Airy, l -JWKB, JWKB on both sides of turning point. This seems complicated, but it leads to a result that will be exceptionally useful! 2. WKB quantization condition: energy levels without wavefunctions! 3. compute dn E /dE (for box normalization — can then convert to any other kind of normalization) 4. trivial solution of Harmonic Oscillator E v = h ω (v+1/2) v = 0, 1, 2… Non-lecture (from pages 6-12 to 6-14) classical ψ a AIRY −1 / 12 2 h m 2 α −1 / 12 ( a x ) −1 / 4 sin 2  2 m 1 / 2 ( a x ) 3 / 2 + π 3 h 2 4 π −1 / 12  2 m −1 / 12 ( x a ) −1 / 4 exp 2  2 m 1 / 2 ( x a ) 3 / 2 forbidden ψ a AIRY = 2 h 2 3 h 2 classical ψ
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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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notes07 - 5.73 Lecture #7 7-1 JWKB QUANTIZATION CONDITION...

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