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08perturbation - Perturbation Theory Only a few QM systems...

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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
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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 dx v a = ψ ψ dx v v l m ml ψ ψ = *
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P460 - perturbation 3
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