**Required math: calculus **

**Required physics: **Schrödinger equation

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.17(b-d).

Here are some more theorems concerning Hermite polynomials, which show up in the solution of the Schrödinger equation for the harmonic oscillator.

The first theorem is that the Hermite polynomials can be obtained from a *generating function*. We’ve seen generating functions in the context of the Laguerre polynomials, which occur in the physics of the hydrogen atom. The derivation of generating functions is something of a black art, and as it requires the use of complex variable theory (in particular, Cauchy’s integral formula) we’ll just accept it without proof for now. The result is

Here is a dummy variable which is used to generate the Taylor series of the exponential on the left. Since the th derivative with respect to of the series eliminates all powers with , retains terms for and reduces the term in to , if we take the th derivative and then set we’re left, magically, with . Taking high order derivatives of the exponential isn’t exactly pretty, of course, but it’s quite marvellous that such a result exists at all.

As an example, we’ll use the generating function to derive the first three polynomials. We get

Starting from the generating function, we can derive two recursion relations for the polynomials. If we take the derivative of 1 with respect to we get

If we replace the exponential on the left by its series expansion, we get

We now pull the usual trick of relabelling the summation index on the first and last sum in order to make the power of the same in all sums. For the first sum, we get

For the sum on the right, we get

Combining these results we get

For , we can equate the coefficients of to get

For the special case of the first sum on the left makes no contribution and the other two terms give us

which, since , gives us , which is correct. We saw when discussing the Rodrigues formula that

so we can use the recursion relation to get the next couple of polynomials:

A second recursion relation can be found by differentiating 1 with respect to rather than . We get

Again, we replace the exponential by the series to get

Relabelling the summation index on the left, we get

Equating coefficients of we get

For example:

### Like this:

Like Loading...

*Related*

Pingback: Perturbing the 3-d harmonic oscillator | Physics pages

WilliamShouldn’t equations 4 and 7 be d/dz not d/dx?

gwrowePost authorFixed now. Thanks.

Pingback: Harmonic oscillator: Hermite polynomials and orthogonality of eigenfunctions | Physics pages

Pingback: Harmonic oscillator: matrix elements using Hermite polynomials | Physics pages

Pingback: Harmonic oscillator – mean position and momentum | Physics pages

Pingback: Harmonic oscillator energies and eigenfunctions derived from the propagator | Physics pages