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 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
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P460 - harmonic oscialltor 3 Harmonic Oscillators • Solve by first looking at large |u| • for smaller |u| assume • Hermite differential equation. Solved in 18th/19th
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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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08harmonic - Harmonic Oscillators F=-kx or V=cx2. Arises...

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