# Hermite Polynomials Assignment

We did not give a precise definition of Hermite polynomials H, (\$) in class, but one precise, albeit indirect, definition is to require the coefficients a; inn,(g) = ajgj=0to satisfy the recursion relationaj+22j+1 -K2(j – n)( + 1) (j+2)j” (+1)(+2)OZCA D-where we have used K = 2n + 1 in the final equality, as well as fixing the highest-power coefficient an to be 2″ and the second-highest-power coefficient an-1 to be 0.

This definition makes it clear that the stationary solutions to the harmonic oscillator should involve H, ().but it is perhaps not a very handy definition. A much more direct definition is to provide an explicit expression:(-1)7-4n!(20)! (9 – 0)!(25) 26,for even n.H (() =n!IM.(-1) 7(20 + 1)! (121 – 0)!(25)24+1, for odd n.Verify the explicit expression above satisfies the indirect definition stated earlier (so that we can be sure that the two definitions agree). Get Science homework help today

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