M Cohen

On the Schrodinger equation with a Gaussian potential

Eigenvalues and approximate eigenfunctions of the Schrodinger equation with an attractive radial Gaussian potential are obtained from a first-order perturbation treatment based on a scale harmonic oscillator model. The bound-state energies are of comparable accuracy to those obtained using high-order perturbation theory by numerical integration.

A unified treatment of Schrodinger's equation for anharmonic and double well potentials

The authors exploit some exact soluble model potentials VN( alpha , beta )=V(A,B)+ lambda x6 in order to extrapolate (as lambda to 0) reliable eigenvalues of the one-dimensional Schrodinger equations with either anharmonic or symmetric double well potentials of the form V(A,B)=1/2Ax2+Bx2 (B>0). Their procedure, which corresponds to low-order Rayleigh-Schrodinger perturbation theory, is found to be competitive with both high-order Pade summation of conventional RSPT and large-scale variational calculations using harmonic oscillators basis functions.

Rayleigh-Schrodinger perturbation theory with a strong perturbation: anharmonic oscillators

The bound state solutions of Schrodingers equation for the anharmonic oscillator potentials V=x2+ lambda x2k (k=2,3, . . . ) have been investigated, using elementary techniques of low-order variational perturbation theory. For the quartic oscillator (k=2) a scaled harmonic potential provides a remarkably accurate model for all lambda . Although this model is slightly less satisfactory for higher-order anharmonicities (k>or=3), the perturbation procedures remain effective, and can be applied successfully provided that higher-order terms are calculated.

Exact time-dependent solutions for a double-well model

Lie algebraic techniques are used to obtain exact solutions of the time-dependent Schrodinger equation for a model double-well potential with an applied, time-dependent, dipole field. The model potential consists of harmonic potentials in x > 0 and x < 0 with an interface region spanning the origin and the theory of the matching of the wavefunctions for the three different regions is examined in detail. The time-dependent solutions are shown to give rise to two independent types of charge transfer arising from a positional change in the wave packet due to the applied field and the change of shape of the wave packet due to interference effects.

Lagrange multipliers and bounds to quantum mechanical properties. II

For pt.I see ibid., vol.4, p.761 (1971). Bounds to overlap integrals between approximate and exact wavefunctions have been derived for arbitrary excited states of quantum mechanical system, using the Lagrange multipliers technique. A new lower bound to the ground state energy which is an improvement over the classical result of Temple has also been derived.

On the Schrodinger equation for the interaction x2+ λx2/(1+gx2)

The lowest even- and odd-parity energy levels of Schrodingers equation for the interaction x2+ lambda x2/(1+gx2) for arbitrary positive values of lambda and g are obtained accurately from a first-order variational perturbation treatment.

An examination of the accuracy of truncated perturbation theory using bounds for the exact energy

A procedure is developed for assessing the accuracy of a truncated perturbation expansion, which leads to upper and lower bounds to some eigenvalue. Improved bounds, which require a lower bound to the overlap of the approximate and exact wavefunctions, are also considered and a lower bound to the overlap is derived. Two examples are presented, using model problems.

An iterative solution of Schrodinger's equation

The energy function E( lambda ) of a quantum system with Hamiltonian H( lambda )=H0+ lambda H1 is determined by an analytical iterative procedure which yields a rich variety of functional forms. The wavefunction iterates include the solutions of Rayleigh-Schrodinger perturbation theory as well as some additional functions, all of which may be obtained sequentially by standard analytical techniques. At each step, an arbitrary ( lambda -dependent) constant may be chosen freely so as to improve the results of low-order iterates, thereby enhancing the convergence of the procedure.

Polynomial perturbation of a hydrogen-like atom

The ground state energy of a hydrogenic atom of nuclear charge Z, perturbed by a polynomial perturbation 2 lambda Zr+2 lambda 2r2, is calculated by means of a variational modification of Rayleigh-Schrodinger perturbation theory, which is effective for all negative lambda .

The perturbed hydrogen atom: some new algebraic results

The authors have employed algebraic methods to calculate the bound-state spectra of a non-relativistic hydrogen atom subjected to a wide class of perturbations. Their procedure exploits the linearity of the complete (perturbed) Hamiltonian in the generators of the SO(2, 2) Lie algebra which follows naturally from the separation of variables in Schrodingers equation in parabolic coordinates. Appropriate transformations then allow the Hamiltonian to be expressed as a linear combination of the compact generators of the two underlying SO(2, 1) algebras. They give some examples for which the bound-state spectra can be obtained completely analytically.