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# notes16 - 5.73 Lecture#16 Last time 12 3 1 V x = 2 kx ax 16...

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16 - 1 5.73 Lecture #16 modified 10/1/02 10:06 AM Perturbation Theory III Last time V( x ) = 1 2 k x 2 + a x 3 cubic anharmonic oscillator algebra with x 3 vs. operator with a , a a x Y x 3 00 ↔ ω & can’t know sign of a from vibrational information alone. [Can know it if rotation- vibration interaction is included.] Morse Oscillator V( x ) = D 1− e α x [ ] 2 * D, α ω , ω x,m * d 3 V dx 3 = 6 a = − 3 h 2 ω 2 α 3 ω x = d 3 V morse dx 3 x =0 * ω x = 2 a 2 h m 3 ω 4 direct from Morse vs. 15 4 a 2 h m 3 ω 4 from pert. theory on 1 2 k x 2 + a x 3 1. 2. Today: 1. Effect of cubic anharmonicity on transition probability orders of pert. theory, convergence [last class: #15-6,7,8]. 2. Use of harmonic oscillator basis sets in wavepacket calculations. 3. What happens when H (0) has degenerate Diagonalize block which contains (near) degeneracies. “Perturbations” — accidental and systematic. 4. 2 coupled non-identical harmonic oscillators: polyads. E n ( 0 ) ’s? ω x = 2 a 2 h m 3 ω 4 from pert. theory (#15- 4) ω x = 15 4 a 2 h m 3 ω 4 same functional form

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16 - 2 5.73 Lecture #16 modified 10/1/02 10:06 AM One reason that the result from second-order perturbation theory applied directly to V(x) = k x 2 /2 + a x 3 and the term-by-term comparison of the power series expansion of the Morse oscillator are not identical is that contributions are neglected from higher derivatives of the Morse potential to the (n + 1/2) 2 term in the energy level expression. In particular E n ( 1 ) = ′′′′ V 0 ( ) x 4 4 ! = 7 / 2 h ω 2 α 4 ω x x 4 24 n x 4 n = h 2 m ω 2 4 n +1 / 2 ( ) 2 + 2 [ ] contributes in first order of perturbation theory to the (n + 1/2) 2 term in E n . E n ( 1 ) = 7 12 ω x n +1 / 2 ( ) 2 + 7 24 ω x Example 2 Compute some property other than Energy (repeat of pages 15-6, 7, 8) need ψ n = ψ n ( 0 ) + ψ n ( 1 ) P n n x n n 2 transition probability: for electric dipole transitions For H - O n n ±1 only x nn+1 2 = h 2 m ω n +1 ( ) for perturbed H-O H (1) = a x 3 ψ n = ψ n ( 0 ) + Σ k H kn ( 1 ) E n ( 0 ) E k ( 0 ) ψ k ( 0 ) ψ n = ψ n ( 0 ) + H nn +3 ( 1 ) −3 h ω ψ n +3 ( 0 ) + H nn +1 ( 1 ) h ω ψ n +1 ( 0 ) + + H nn −1 ( 1 ) h ω ψ n −1 ( 0 ) + H nn −3 ( 1 ) 3 h ω ψ n −3 ( 0 )
16 - 3 5.73 Lecture #16 modified 10/1/02 10:06 AM n+4 n+2 n+1 n n–1 n–2 n–4 n n n+3 n+1 n–1 n–3 n+7 , n+5, n+4, n+3, n+1 n+5, n+3, n+2, n+1, n–3 n+4, n+2, n+1, n, n–2 n+3, n+1, n, n–1, n–3 n+2, n, n–1, n–2, n–4 n+1, n–1, n–2, n–3, n–5 n-1, n-3, n-4, n-5, n-7 effect of x anharmonic final state ψ n ( 0 ) + ψ n ( 1 ) initial state ψ n ( 0 ) Many paths which interfere constructively and destructively in x n n 2 n = n + 7 ,n + 5 ,n + 4 ,n + 3 ,n + 2 ,n +1 ,n,n – 1 ,n − 2 ,n − 3 ,n − 4 ,n − 5 ,n − 7 only paths for H-O!

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