If we start with the ground state we would expect that we can’t go any lower,
(9.17) |
This can of course be checked explicitly,
Quiz Can you show that using the operators ?
Once we know that , repeated application of Eq. ( 9.15 ) shows that , which we know to be correct from our previous treatment.
Actually, once we know the ground state, we can now easily determine all the Hermite polynomials up to a normalisation constant:
Indeed .
From math books we can learn that the standard definition of the Hermite polynomials corresponds to
(9.20) |
We thus learn and .
Question: Prove this last relation.