The Eigenvector-Eigenvalue Problem

The eigenvector-eigenvalue problem is one with the form of,

\[ A f = \lambda f \]

• \(A\) : matrix or operator applied to the eigenfunction

• \(f\) : eigenfunction or eigenvector

• \(\lambda\) : the eigenvalue, a constant

The goal is often to find both the eigenfunction (e.g., wavefunction to the time independent Schrodinger equation) and the eigenvalue (e.g., energy corresponding to the wavefunction)

In quantum mechanics we only deal with hermitian operators which may be represented as an actual operator or a matrix. With the hermitian operator there are two types of eigenfunctions discrete and continuous eigenfunctions.

Discrete Eigenfunctions

The discrete eigenfunctions are physically realizable states.

• Eigenvalues are real

• Eigenfunctions are orthogonal to other eigenfunctions if their eigenvalue are different.

• Eigenfunctions with same eigenvalues are considered to be degenerate states.