08perturbation - P460 - perturbation 1 Perturbation Theory...

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Unformatted text preview: P460 - perturbation 1 Perturbation Theory • Only a few QM systems that can be solved exactly: Hydrogen Atom(simplified), harmonic oscillator, infinite+finite well • solve using perturbation theory which starts from a known solution and makes successive approx- imations • start with time independent. V’(x)=V(x)+v(x) • V(x) has solutions to the S.E. and so known eigenvalues and eigenfunctions • let perturbation v(x) be small compared to V(x) ' ) ( / V of solutions a x V of ions eigenfunct l l nl n l ⇒ = ⇒ ∑ ψ ψ ψ As ψ l form complete set of states (linear algebra) Sometimes Einstein convention used. Implied sum if 2 of same index P460 - perturbation 2 Plug into Schrod. Eq. • know solutions for V • use orthogonality • multiply each side by wave function* and integrate • matrix element of potential v is defined: ∑ ∑ ∑ = + + = + + − − l nl n l nl l nl mdx d n n n mdx d a E v a a V E v V n ψ ψ ψ ψ ψ ψ / 2 / / / 2 ] [ ) ( 2 2 2 2 / 2 2 h h ∑ ∑ ∑ = + l nl n l nl l l nl a E v a E a ψ ψ ψ / ml l m dx δ ψ ψ = ∫ * ) ( / * m n nm l m nl E E a...
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This note was uploaded on 10/02/2009 for the course PHYS 460 taught by Professor Johnson,c during the Spring '08 term at Northern Illinois University.

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08perturbation - P460 - perturbation 1 Perturbation Theory...

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