Supersymmetric Quantum Mechanics, Excited State Energies and Wave Functions, and the Rayleigh−Ritz Variational Principle: A Proof of Principle Study

In addition to ground state wave functions and energies, excited states and their energies are also obtained in a standard Rayleigh−Ritz variational calculation. However, their accuracy is generally much lower. Using the super-symmetric (SUSY) form of quantum mechanics, we show that better accuracy and more rapid convergence can be obtained by taking advantage of calculations of the ground states of higher sector SUSY Hamiltonians, followed by application of the SUSY “charge operators”. Our proof of principle study uses a general family of one-dimensional anharmonic oscillator models. We first obtain the exact, analytic ground states for a general family of anharmonic systems. We give the general, factorized form of the Hamiltonian for the hierarchy that arises in SUSY theory. The “charge” operators can then be used to convert states among the sectors. We illustrate the approach with two specific anharmonic oscillator models. Using the ground state of the second sector Hamiltonian, we show that the corresponding excited state energies and wave functions of the first sector are accurately obtained by applying the charge operators, using significantly smaller basis sets than are required in a standard variational approach applied to the original Schrödinger equation. This is a consequence of the higher accuracy of the Rayleigh−Ritz variational method when applied for ground states.