Analysis of Energies and Nodes through a First-Order Schrodinger Equation

For most potentials, the Schrödinger equation does not admit exact solutions. However, some exact bounds are possible. We transform the one-dimensional time-independent Schrodinger equation into a well-behaved first order differential equation. Many properties of energy eigenfunctions are seen geometrically through the direction field of the transformed Schrodinger equation. Using quasi-eigenfunctions, we prove a generalization of the node theorem which yields the number of nodes of the quasi-eigenfunction based on the position of the quasi-eigenfunction's energy in the system's energy spectrum. We use these results to prove bounds on the energies of perturbed potentials. Unlike standard perturbation theory results, this treatment remains valid even for large perturbations, allowing us to address problems where perturbation theory is not applicable. This work provides a novel method to analyze quantum mechanics problems that is complementary to well established perturbation theory, variational principle, and WKB techniques.