08perturbation2 - Time Dependent Perturbation Theory Many...

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P460 - perturbation 2 1 Time Dependent Perturbation Theory • Many possible potentials. Consider one where V’(x,t)=V(x)+v(x,t) • V(x) has solutions to the S.E. and so known eigenvalues and eigenfunctions • let perturbation v(x,t) be small compared to V(x) • examples:finite square well plus a pulse that starts at t=0 or atoms in an oscillating electric field • write wavefunction in terms of known eigenfunctions but allow the coefficients (c n ) vary with time b fraction with any eigenvalue (say energy) changes with time ) ( ) , ( ) ( / x e t x x V of ions eigenfunct n t iE n n n ψ h = Ψ ' ) ( ) , ( ' V of solutions t c t x n n n Ψ = Ψ
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P460 - perturbation 2 2 Time Dependent • start from unperturbed with |n> time independent • add on time-dependent perturbation and try to find new, time-dependent wavefunctions . λ keeps track of the order . Gives Schrodinger Eq.: • write out ψ in terms of eigenfunctions |n> • substitute into (2) and eliminate part that is (1) ) 1 ( 0 0 n E n H n = ) 2 ( ) ( ) ( ) ( )) ( ( 0 t dt d i t H t t V H ψ λ h = = + n e t c t iE n n h / 0 ) ( = [ ] 0 ) ( / / 0 / 0 / 0 / 0 0 0 0 0 = + = + + = n e Vc dt dc i n e c V E n e c V H n e c E dt dc i n e c dt d i t iE n n t iE n n t iE n t iE n n n t iE n n n n n n h h h h h h h h
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P460 - perturbation 2 3 Time Dependent • take dot product with <m| • this is exact. How you solve depends on the potential and initial conditions • an example. start out at t<0 in state k. “turn on” perturbation at t=0. See what the values of c m are at a later time • reminder. c m gives probability to be in state m • for first order, just do transitions from k b m. ignore k b n b m and higher (as the assumption that all are in state k won’t hold for t>0) n V m e c dt dc i n e Vc dt dc i m e t E E i n m t iE n n t iE n m n m h h h h h / ) ( / / 0 0 0 0 0 1 = = λ 2 2 | ) ( | | | ) ( t c m t P m m = = ψ
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P460 - perturbation 2 4 Time Dependent – example try to solve. assume in particular state k at t=0
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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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08perturbation2 - Time Dependent Perturbation Theory Many...

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