{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}