2016/07/13 by Zafar Ahmed, Ahmed, Zafar, Mohammad Irfan +5
Physics and Astronomy · #FOS: Physical sciences #General Physics (physics.gen-ph) #physics.gen-ph
paper · pdf · doi:10.48550/arxiv.1608.01543
10 pages, 3 figures and 1 Table
arxiv created 2016/08/05 · arxiv updated 2016/08/08
We draw attention to the fact that a Hermitian matrix is always diagonalizable and has real discrete spectrum whereas the Hermitian Schrödinger Hamiltonian: H=p2/2μ+V(x), may not be so. For instance when V(x)=x, x3, -x2, 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 Hψn=En ψn into a matrix and get eigenvalues En by diagonalizing the matrix. We show, that the thus obtained En 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.