Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Chapter 2, Post 17(a).
There are several theorems concerning Hermite polynomials, which show up in the solution of the Schrödinger equation for the harmonic oscillator.
First, we’ll look at the Rodrigues formula (which is a different formula from the Rodrigues formula for Legendre polynomials).
Suppose we start with and take its derivative. We have
We can now take the derivative of the second equation times and use Leibniz’s formula for the th derivative of a product, which is
Since any derivative of higher than the first gives zero, we have
Applying this to the original equation, we get
Defining yet another variable we get (the factor of is inserted to make things come out right at the other end):
Finally, defining , we have
Substituting this into 1 we get, after dividing out the common factor of :
This last equation is the same as that obtained from the Schrödinger equation earlier (with different variable names):
We can see by comparing the two forms of the equation that a solution to the latter is
Since this is a solution it must be a multiple of the Hermite polynomial. To see that it is actually the Hermite polynomial itself, consider the derivative term. Each derivative of will have a term multiplying the previous derivative by , so the term with the highest power of in the th derivative will be . We now see why the factor of was introduced earlier: by the usual convention, the coefficient of the highest power of a Hermite polynomial is , which is what we obtain from the formula above. Thus the Rodrigues formula for Hermite polynomials is
We can apply this formula directly to get the first few polynomials. We get