Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Chapter 2, Post 16.
In the solution of the Schrödinger equation for the harmonic oscillator, we found that the wave function can be expressed as a power series:
where was introduced as a shorthand variable:
and the coefficients satisfy the recursion relation
By requiring the recursion relation to terminate at various values of we can generate the polynomials for the various energy states, which turn out to be the Hermite polynomials.
For example, if we take the highest value of to be 5, then we must have
To get the coefficients, we can start with (since all even terms must be zero if we want ) and . Then we get
so , . The Hermite polynomial is
where is a constant that is set by convention to make the coefficients satisfy some specified rule.
If we require the coefficient of the highest power to be we can multiply this polynomial by 120 to get
For , all the odd terms are zero, and we require , so we get
Taking and , we get , , . Requiring the coefficient of to be we get