notes15 - 5.73 Lecture #15 15 - 1 Perturbation Theory II...

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15 - 1 5.73 Lecture #15 modified 9/30/02 10:12 AM Perturbation Theory II (See CTDL 1095-1104, 1110-1119) Last time: H ( 0 ) ψ n ( 0 ) = E n ( 0 ) n ( 0 ) H ( 0 ) is diagonal n (0) {} ,E n ( 0 ) are basis functions and zero - order energies E n (1) = H nn expectation value of perturbation operator E n (2) = Σ k H nk 2 E n E k sum excludes k = n matrix element vs. energy denominator E n = E n + E n + E n n = Σ k H nk E n E k k sum excludes k = n 1st index mixing coefficient, order sorting parameter, convergence criterion Today: 1. cubic anharmonic perturbation x 3 vs. a , a a x 3 ω x and Y 00 contributions 2. nonlecture Morse oscillator pert. theory for a x 3 3. transition probabilities — orders and convergence of p.t. Mechanical and electronic anharmonicities.
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15 - 2 5.73 Lecture #15 modified 9/30/02 10:12 AM Example 1. H = p 2 2 m + 1 2 k x 2 + a x 3 H (0) H (1) x V(x) (a < 0) unphysical need matrix elements of x 3 one (longer) way x i l 3 = x ij x jk x k l j,k 4 different selection rules: l – i = 3, 1, –1, –3 l – i = 3 i i +1, i + 1 i + 2, i + 2 i + 3 l – i = 1 i i + 1, i + 1 i + 2, i + 2 i + 1 i i – 1, i – 1 i, i i + 1 i i + 1, i + 1 i, i i + 1 i +1 () i +2 i +3 [] 1 / 2 i i i 1 / 2 + i i i 1 / 2 + i i i 1 / 2 algebraically complicated other (shorter) alternative: a , a , and a a xx a a aa 3 32 3 12 3 3 2 2 = = + = + hh h mm m ωω ω / ~ / /† / a + a 3 = a 3 + a aa + aa a + aaa + aa a + a aa + a a a + a 3 four terms, four different selection rules. one path There are three 3-step paths from i to i + 1. Add them.
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15 - 3 5.73 Lecture #15 modified 9/30/02 10:12 AM Use simple a , a algebra to work out all matrix elements and selection rules by inspection. recall: a n = n +1 () 1 / 2 n , a n = n 1 / 2 n −1 , a a n = nn a , a [] =1 aa =1+ a a prescription for permuting a thru a n = 3 a n −3 ,n 3 = n −2 n n 1 / 2 n =+3 a n +3 3 = n n +2 n 1 / 2 n =−1 a aa + aa a + aaa n goal is to rearrange each product so that it has number operator at right a aa = aa a a aaa = aa a + a aa
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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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notes15 - 5.73 Lecture #15 15 - 1 Perturbation Theory II...

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