## SHM p.8

### Raising and Lowering Operators

Any set of orthogonal polynomials must satisfy an equation like (why?):
*xP*_{n}(*x*) =
*a*_{n}P_{n+1}(*x*) +
*b*_{n}P_{n}(*x*) +
*c*_{n}P_{n-1}(*x*)

In the case of polynomials with definite even/odd parity,
*b*_{n}=0 (why?). For Hermite polynomials we have:

*x H*_{n}(*x*) =
½ *H*_{n+1}(*x*) +
*n H*_{n-1}(*x*)

All the classical orthogonal polynomials satisfy equations like:

*a*_{n}(*x*) *P'*_{n}(*x*) =
*b*_{n}(*x*) *P*_{n}(*x*) +
*c*_{n}(*x*) *P*_{n-1}(*x*)

For Hermite polynomials we have:

*H'*_{n}(*x*) =
2*n H*_{n-1}(*x*)

Since

we can relate the results of simple operations on an eigenfunction
in terms of other eigenfunctions

By themselves these results are useful, e.g.,
to calculate integrals like: <*n*|*x'*^{2}|*n*>
or <*n*|*p*^{2}|*n*>, but they find their
most common use by combining the results into
*raising* and
*lowering*
operators:

We note the following properties of the *raising* and
*lowering* operators:

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