In quantum mechanics, if any eigenstate is -fold degenerate, there are an infinite number of choices for the orthogonal eigenfunctions. The simplest possible example is the free particle in one dimension. Every energy level is twofold degenerate. This corresponds to the physical fact that particles moving in opposite directions have the same kinetic energy. The Schrödinger equation has two linearly independent eigenfunctions. A common choice takes . These functions are delta function-normalized, such that , and are also eigenfunctions of linear momentum , with the eigenvalues .[more]
In this Demonstration, you can explore any of an infinite number of orthonormalized degenerate pairs of eigenfunctions, which can be represented by and , where . Both real and imaginary parts of each wavefunction are plotted.[less]
Snapshots 1 and 2: varying values of
Snapshot 3: for , the eigenfunctions are proportional to and
Reference: S. M. Blinder, Introduction to Quantum Mechanics, Amsterdam: Elsevier, 2004 pp. 31–32.