Franz Schwabl Quantum Mechanics 4th Edit · Problem 3.1
The Uncertainty Relation
Physics — Quantum Mechanics · hermite-polynomials
Problem
The Hermite polynomials Hn(x) are defined in (3.25).
(a) Show that e−t2+2tx is a generating function of the Hermite polynomials, i.e.,
e−t2+2tx=n=0∑∞n!tnHn(x).
(b) By taking the derivative of this expression, show the validity of the recursion relations
Hn′(x)=2nHn−1(x)
and
Hn+1(x)=2xHn(x)−2nHn−1(x).
(c) Prove the completeness relation (3.30). Hint: Express Hn(x) and Hn(x′) in terms of (3.25) and represent the Gaussian by its Fourier transform.
(d) Show that the Hermite polynomials and hence the wave functions ψn have n nodes (n simple real zeros) for finite x. Hint: proof by induction, using Rolle's theorem.