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# 08harmonic - Harmonic Oscillators F=-kx or V=cx2 Arises...

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P460 - harmonic oscialltor 1 Harmonic Oscillators V F=-kx or V=cx 2 . Arises often as first approximation for the minimum of a potential well Solve directly through “calculus” (analytical) Solve using group-theory like methods from relationship between x and p (algebraic) Classically F=ma and m C m C dt x d t x x π ν πν 2 1 2 sin 0 2 2 = = = + Sch.Eq. 0 ) ( 2 2 2 2 2 2 2 2 2 2 2 2 = + = = = = + ψ α β α ψ ψ α β ψ ν π ψ u x u define E x du d mE m C mdx d h h h

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P460 - harmonic oscialltor 2 Harmonic Oscillators-Guess V Can use our solution to finite well to guess at a solution Know lowest energy is 1-node, second is 2-node, etc. Know will be Parity eigenstates (odd, even functions) Could try to match at boundary but turns out in this case can solve the diff.eq. for all x (as no abrupt changes in V(x) ψ
P460 - harmonic oscialltor 3 Harmonic Oscillators Solve by first looking at large |u| for smaller |u| assume Hermite differential equation. Solved in 18th/19th century. Its constraints lead to energy eigenvalues

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