The harmonic oscillator potential very rich physics with interesting classical/quantum connections exploit oddness/evenness again it will be first tackled analytically; later algebraically all states are bound states and satisfy E > 0 to make most progress, we will ‘split off’ long-distance behavior, which is, as we will show, exponential decay for x  ± ∞

The harmonic oscillator potential recast to be non-divergent (this is the ‘square-integrable’ requirement)at  ± ∞, we must insist B = 0  is a pure gaussian in  (and therefore in x) asymptotically define h(  ) via

The harmonic oscillator power series standard method for ODE’s with powers is the power series, so try

The HO power series is even or odd there will be an ‘even’ series and an ‘odd’ one establish (from boundary conditions) a 0 and a 1  all else follows however, what is the ‘large j’ behavior of the series? This will control the way in which h(  ) diverges for large x K is constant; for j is large we have approximately

The HO power series must terminate: energy quantized this is nasty growth for   ±∞; even the gaussian dropoff is too weak to make  square-integrable our only recourse is to terminate the series at n: a n+2 = 0 therefore the even power series for h takes on the form how would this work for odd n? [difference in detail only] CONLUSIONS: energies are quantized and differ by ћ  ; there is a ground-state energy for n = 0

consider any potential function V(x) that possesses a minimum ‘bowl’ at some position x 0 Taylor expand V(x) about that position, to second order: How ubiquitous is the HO? Thus we see that apart from the additive and non-important constant, the potential energy takes on the form of an HO’s, and the ‘curvature’ at the minimum is precisely the spring constant!!!

The HO wavefunctions are alternately even, odd… now the recursion relation takes over for the higher even/odd states how would this work for odd n? [difference in detail only] CONLUSIONS: energies are quantized and differ by ћ  ; there is a ground-state energy for n = 0

note the manifest evenness or oddness ‘Gaussian’ integrals are very very handy in this game example: normalize the ground state [a perfect Gaussian] The HO wavefunctions as Hermite polynomials Hermite polynomials {H n (  )} are ‘scaled’ so that the highest power of the argument is 2 n then, when normalized in the usual way, the HO {  n (  )} are the first few are Hermite polynomials {H n (  )} are

a couple of interesting ways to generate the Hermites: The Hermite polynomials for their own sake again, Hermite polynomials {H n (  )} are ‘scaled’ so that the highest power of the argument is 2 n they are just one of a plethora of ‘special function sets’ in mathematical physics Bessel (first kind, second kind: Neumann), Legendre, associated Legendre, gamma, zeta… they are most useful if orthogonal their ‘scale’ is set in a context-dependent way since they are often solutions to linear ODEs, the amplitude of the solution to a particular problem is established by some kind of boundary conditions