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Unformatted text preview: 0307Jan2011 Chemistry 21b – Molecular Structure & Spectroscopy Week # 1 Supplemental Notes – Perturbation Theory Perturbation Theory in Quantum Mechanics Before we launch into a detailed investigation of the ways in which molecules, liquids, and solids interact with electromagnetic radiation, beyond the nature of the eigenstates of these systems we will need to flesh out an additional component to our quantum mechanical toolkit – perturbation theory. Ultimately, for spectroscopy we will need to utilize what is called timedependent perturbation theory, but for starters in these supplementary notes we will look at timeindependent perturbation theory to build up some necessary mathematical formalisms. Timeindependent perturbation theory comes in two flavors, nondegenerate (that is, all of the eigenvalues are distinct) and degenerate (that is, a system in which some of the eigenvalues may be identical) perturbation theory. In all cases, the basic idea is to split the total Hamiltonian into two parts, or ˆ H = ˆ H + ˆ H ′ , where ˆ H is called the zerothorder Hamiltonian and ˆ H ′ is the operator for the perturbation. For the approach to work well, it is best if the eigenvalues of ˆ H are known. Some examples of important problems that can be treated with perturbation theory include L · S coupling in lowZ atoms, anharmonic oscillators, and the Stark and Zeeman effects (That is, the response of atoms and molecules to static electric and magnetic external fields.). The timedependent interactions of atoms and molecules with electromagnetic fields is another important example that we will spend most of the course looking at. In order for perturbation theory to work well, the effects of the perturbing term ˆ H ′ must be, in some sense, small compared to that of ˆ H . Put another way, we will assume that the eigenstates and eigenenergies of the total Hamiltonian, let’s call them { ϕ n } and { E n } , are only slightly different from those of the unperturbed Hamiltonian, ˆ H , which we’ll call { ϕ (0) n } and { E (0) n } . Provided the effects of ˆ H ′ are indeed small, we can then introduce the terms Δ ϕ n and Δ E n as small corrections to the zerothorder energies and wavefunctions, or ϕ n = ϕ (0) n + Δ ϕ n E n = E (0) n + Δ E n . To keep the smallness of ˆ H ′ explicitly in mind, many textbooks rewrite the first equation above as containing the term λ ˆ H ′ , where λ is an infinitesimal parameter and is introduced for “bookkeeping” purposes. Thus, the equation we must solve becomes ( ˆ H + λ ˆ H ′ ) ϕ n = E n ϕ n . (1 . 1) The Perturbation Expansion Again, we assume that the eigenstates and eigenenergies of ˆ H are known, and that the eigenstates form a complete, orthonormal basis set which spans the Hilbert space of ˆ H ....
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