Introduction to Quantum Mechanics · Problem 3.7

Formalism — David J. Griffiths

Physics — Quantum Mechanics · eigenvalues_eigenfunctions

Problem

(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) = \exp(x)

and

g(x) = \exp(-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)