P460 - harmonic oscialltor1 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 Sch.Eq.

P460 - harmonic oscialltor2 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 Partity 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) 

P460 - harmonic oscialltor3 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 oscialltor4 Hermite Equation put H, dH/du, d 2 H/du 2 into Hermite Eq. Rearrange so that you have for this to hold requires that A=B=C=….=0 gives

P460 - harmonic oscialltor5 Hermite Equation Get recursion relationship if any a l =0 then the series can end (higher terms are 0). Gives eigenvalues easiest if define odd or even types Examples

P460 - harmonic oscialltor6 Hermite Equation - wavefunctions Use recursion relationship to form eigenfunctions. Compare to first 2 for infinite well First Second Third

P460 - harmonic oscialltor7 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 oscialltor8 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: