Science Homework Help
Please show how to these hermite polynomials with recursion
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; innH,(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-powercoefficient an to be 2″ and the second-highest-power coefficient an-1 to be 0. This definitionmakes 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 providean 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 thatwe can be sure that the two definitions agree).
