12_variation_perturbation_matrixhq

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Unformatted text preview: Time Independent given known (unperturbed) solutions with superscripted zeroes (1) (2) Solve a new problem where new Hamiltonian is small change ( ) from old one Find the perturbed energy and wavefunctions of As → 0 1, this goes from unperturbed to perturbed (3) Expand , , and in ⋯ (4) λ ⋯ (5) λ ⋯ (6) plug eqs. (4-6) into (3) 21 10/7/2013 the Schrödinger equation is ⋯ λ λ ⋯ ⋯ λ ⋯ (7) multiplying-out, collecting terms in ⋯ since (8) 0 is arbitrary, each power of is zero separately giving a set of equations (9) (10) (11) (11) Write eq. (10) in Bra-Ket notation (12) + left multiply by ˆ H ( 0) n (0) E (0) n (0) + (13) + (14) adjoint ˆ n (0) H (0) n (0) E (0) (15) (16) (17) ∗ first-order correction to energy 22 10/7/2013 Particle-in-a-Box is the zero order solution for a perturbative treatment of the Particle-in-aRamped-Box Particle-in-a-Ramped-Box (Effect of a DC Electric Field) Particle-in-a-Box Vi 0.5 i .2 e v 0.4 Energy (Hartree) Energy (Hartree) . ev 0.5 Vi i ....
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