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Harmonic Oscillators

Harmonic Oscillators . 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). V. Classically F=ma and. Sch.Eq.

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Harmonic Oscillators

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  1. Harmonic Oscillators • F=-kx or V=cx2. 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) V Classically F=ma and Sch.Eq. P460 - harmonic oscialltor

  2. Harmonic Oscillators-Guess • 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) V y 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 • Solve with Series Solution technique P460 - harmonic oscialltor

  4. Hermite Equation • put H, dH/du, d2H/du2 into Hermite Eq. • Rearrange so that you have • for this to hold requires that A=B=C=….=0 gives P460 - harmonic oscialltor

  5. Hermite Equation • Get recursion relationship • if any al=0 then the series can end (higher terms are 0). Gives eigenvalues • easiest if define odd or even types Examples P460 - harmonic oscialltor

  6. Hermite Equation - wavefunctions • Use recursion relationship to form eigenfunctions. Compare to first 2 for infinite well • First • Second • Third P460 - harmonic oscialltor

  7. H.O. Example • A particle starts in the state: • Split among eigenstates • From t=0 • what is the expectation value of the energy? P460 - harmonic oscialltor

  8. H.O. Example 2 • A particle starts in the state: • At some later time T the wave function has evolved to a new function. determine T • solve by inspection. Need: P460 - harmonic oscialltor

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