Hermitian Hamiltonians: Matrix versus Schr${\"o}$dinger's

TL;DR

This paper discusses the differences between Hermitian matrices and Hermitian Schrödinger Hamiltonians, highlighting that the latter may lack real discrete spectra, and proposes a matrix diagonalization approach for spectral analysis when applicable.

Contribution

It clarifies the distinction between matrix and Schrödinger Hermitian operators and suggests a practical matrix diagonalization method for spectral computation in quantum systems.

Findings

Hermitian matrices are always diagonalizable with real spectra.

Hermitian Schrödinger Hamiltonians may lack real discrete eigenvalues.

Matrix diagonalization can accurately find spectra if no scattering states are present.

Abstract

We draw attention to the fact that a Hermitian matrix is always diagonalizable and has real discrete spectrum whereas the Hermitian Schr{\"o}dinger Hamiltonian: $H=p^2/2\mu+V(x)$, may not be so. For instance when $V(x)=x, x^3, -x^2$, $H$ does not have even one real discrete eigenvalue. Textbooks do not highlight this distinction. However, if $H$ has real discrete spectrum, by virtue of the expansion theorem, one can convert the eigenvalue problem $H\psi_n=E_n \psi_n$ into a matrix and get eigenvalues $E_n$ by diagonalizing the matrix. We show, that the thus obtained $E_n$ could be accurate, provided $H$ is devoid of scattering states. We suggest that this could be a simple and apt way to introduce the method of Linear Combination of Atomic Orbitals (LCAO) for finding the spectra of molecules. In textbooks, usually the method of matrix-diagonalization appears meagerly as a degenerate perturbation theory for more than one dimensions.