notes15

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

This preview shows pages 1–4. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online