12_lecnotes_rwf

12_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Robert Field 12 – 1 5.74 RWF Lecture #12 Quasi-degenerate Perturbation Theory. Strong and Weak Coupling Limits Reading : Chapter 9.3, The Spectra and Dynamics of Diatomic Molecules , H. Lefebvre-Brion and R. Field, 2 nd Ed., Academic Press, 2004. Last time: Non-degenerate perturbation theory for interaction between quasi-eigenstates with finite level-width If we allow H (0) to have complex energies along the diagonal, we have a basis for calculating how quasi-eigenstates share the property of decay rate. Illustration that sharing of decay rate (NOT LIFETIME) is just like sharing of Zeeman tuning rate () + ( ) * key is to use perturbation theory to obtain ψ = ψ i 0 ψ i 1 . i * energy denominator is complex - rationalize and get corrections to ε and Γ Problem of orthogonality — solved by biorthogonal basis set. Today: 1. biorthogonality completeness 2. 2 × 2 complex H 3.± limiting cases strong & weak coupling limits 4. doorway state “dissolves in bath” 5. quantum beats . Biorthogonality When H is real everywhere except along the main diagonal H i = E i i then it must be true that * * i ˜ i ˜ H = E i but then we have i i i ˜ = ( a 1 a N ) i a 1 i = M a i N a 1 i * i ˜ = M a i N *
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5.74 RWF Lecture #12 12 – 2 and therefore i 2 ˜ = () a which is not necessarily 1 and can even be complex. ii t Γ / h ! Note that Ψ ( Q , t ) cannot remain normalized to 1 at all t because P( t ) = e So we are stuck with two awkwardnesses about normalization: * cannot insist on normalization to 1 at all t * usual normalization integral can give a complex number We handle this by expressing the ortho-normalization condition as ˜ j i ij ˜ j j −Γ j h (the factor ˜ jj in the denominator cancels the e t decay of probability) and completeness as ˜ . 1 = ˜ j j j non-zero only when j = i Note that ˜ i i = = i 1 ˜ j ˜ ij ˜ j i ˜ 1 = = i ˜ ˜ j j j as expected, and, for a member of a different basis set ˜ jK 1 K = j . ˜ j j j
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5.74 RWF Lecture #12 12 – 3 It turns out that all of the results for E i and ψ i from nondegenerate perturbation theory come out as / 12 ˜ expected. But it is important to remember that it is always necessary to include the factor of [ ] to renormalize i .
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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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12_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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