If we start with the ground state we would expect that we can't go any lower,
| (9.17) |
| = | |||
| = | |||
| = | 0 | (9.18) |
Quiz Can you show that
= 1/2 using the operators
?
Once we know that
= 1/2, repeated application of Eq.
(9.15) shows that
= n + 1/2, 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:
| u1(y) | a |
||
| = | |||
| = | |||
| = | (9.19) |
From math books we can learn that the standard definition of the Hermite polynomials corresponds to
| Hn(y)e-y2/2 = ( |
(9.20) |
Question: Prove this last relation.