On the Nonorthogonality of Generalized Momentum Eigenfunctions in Quantum Mechanics

The orthogonality requirement usually specified for the eigenfunctions φ of a quantum‐mechanical operator P which has a continuous spectrum of eigenvalues λ, is $\mathrm{\int \phi *(\lambda )\phi (\lambda \prime )\hspace{0.17em}d\lambda =\delta (\lambda -\lambda \prime ).}$ The derivation of this result is examined; it is seen to be merely a sufficient, but not a necessary condition for the consistent expansion of a bound‐state wavefunction ψ in the form $\mathrm{\int \hspace{0.17em}a(\lambda )\phi (\lambda )\hspace{0.17em}d\lambda .}$ The situation when P is a generalized momentum operator is investigated in detail. Such operators were discussed in the preceding paper. It is shown that any two of the corresponding momentum eigenfunctions are not usually orthogonal. Nevertheless, with the help of Fourier analysis, a consistent expansion $\mathrm{\psi =\int \hspace{0.17em}a(\lambda )\phi (\lambda )\hspace{0.17em}d\lambda }$ is established. The theory is illustrated with the familiar examples of a ``particle in a box'' and a ground‐state hydrogen atom. When a space coordinate has a finite range, a quantum condition can be imposed on the generalized momentum eigenvalues so that the momentum eigenfunctions form a complete discrete orthogonal set. We attempt to justify the belief that such quantization is not essential unless it is necessary to ensure single‐valued eigenfunctions.