Problem 3.7 – Griffith’s Intro to QM

Problem 3.7

(a) Suppose that $f(x)$ and $g(x)$ are two eigenfunctions of an operator $\hat{Q}$ , with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of $\hat{Q}$ , with eigenvalue q.
(b) Check that $f(x) = e^x$ and $g(x)=e^{-x}$ are eigenfunctions of the operator $d^2/dx^2$ , with the same eigenvalue. Construct two linear combinations of f and g that are orthogonal eigenfunctions on the interval (-1, 1).