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; in
H,(g) = ajg
to satisfy the recursion relation
2(j – n)
( + 1) (j+2)
where we have used K = 2n + 1 in the final equality, as well as fixing the highest-power
coefficient an to be 2quot; 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:
(20)! (9 – 0)!
for even n.
H (() =
(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).Science