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 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 oscialltor3 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 oscialltor4 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 Wave functions from recursion relations

P460 - harmonic oscialltor5 H.O. - algebraic/group theory See Griffiths Sect write down the Hamiltonian in terms of p,x operators try to factor but p and x do not commute. Explore some relationships, Define step-up and step-down operators

P460 - harmonic oscialltor6 H.O.- algebraic/group theory Look at [x,p] with this by substitution get and Sch. Eq. Can be rewritten in one of two ways Look for different wave functions which satisfy S.E.

P460 - harmonic oscialltor7 H.O.- algebraic/group theory Start with eigenfunction with eigenvalue E so these two new functions are also eigenfunctions of H with different energy eigenvalues a + is step-up operator: moves up to next level a - is step-down operator: goes to lower energy.a + (a +  a +    a -  a - (a - 

P460 - harmonic oscialltor8 H.O.- algebraic/group theory Can prove this uses a similar proof can be done for step-down can raise and lower wave functions. But there is a lowest energy level…..

P460 - harmonic oscialltor9 H.O. eigenfunctions For lowest energy level can’t “step-down” easy differential equation to solve can determine energy eigenvalue step-up operator then gives energy and wave functions for states n=1,2,3…..

P460 - harmonic oscialltor10 H.O. Example Griffiths problem Compute,,, and uncertainty relationship for ground state. Could do just by integrating. Instead using step-up and step-down operator. With this But consider

P460 - harmonic oscialltor11 H.O. Example Griffiths problem Can show using normalization (see Griffiths) using “mixed” terms

P460 - harmonic oscialltor12 H.O. Example 2 Griffiths problem A particle starts in the state: what is the expectation value of the energy? At some later time T the wave function has evolved to a new function. determine T

P460 - harmonic oscialltor13 H.O. Example 2 Griffiths problem 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: