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European Journal of Physics

Use of the Schrodinger-Coulomb Green function in the evaluation of the quadratic Stark effect

Gregory S. Adkins, R F Hood

Published Jan 1, 1989• 7 citations
Publisher

Abstract

The Schrodinger-Coulomb Green function is discussed as a tool for evaluating terms in bound-state perturbation theory. As an example the energy shift of the ground state of hydrogen in a uniform electric field is considered. The Green function is used in an evaluation of the second-order perturbation theory expression for this energy shift. For comparison a direct numerical approach to the evaluation of this energy shift is also described. The Green function approach is simpler and more elegant, and leads to an exact result.

DOI
10.1088/0143-0807/10/1/013

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The Schrodinger-Coulomb Green function is discussed as a tool for evaluating terms in bound-state perturbation theory. As an example the energy shift of the ground state of hydrogen in a uniform electric field is considered. The Green function is used in an evaluation of the second-order perturbation theory expression for this energy shift. For comparison a direct numerical approach to the evaluation of this energy shift is also described. The Green function approach is simpler and more elegant, and leads to an exact result.
The response of a hydrogen atom to a perturbative electric field is presented from a classical viewpoint. We find classical energy shifts in the hydrogen atom in situations that correspond to both the linear and quadratic Stark effects. We then compare these energy shifts to the exact quantum-mechanical values for the Stark effect. We find that the linear Stark effect can be accurately reproduced classically (given the correct initial conditions). In contrast, quantum-mechanical results for the quadratic Stark effect cannot be accurately reproduced classically, except in the correspondence principle limit of large principal quantum numbers n.
In this paper, we present an efficient alternative method for the evaluation of the three-photon transition matrix element in the dipole approximation from the ground state to bound states in atomic hydrogen. This method is a variation of the Dalgarno–Lewis method for the treatment of the second-order Stark effect in the hydrogen atom. In this approach, the infinite double sum over the complete set of states including the continuum states present in the third-order perturbation theory result is treated exactly. The closed analytical expression obtained for the matrix element, as a function of incident photon energy, clearly displays all singularities present in the original third-order perturbation theory result.
The second-order perturbative Stark effect on the ground state of hydrogen is a typical example presented in many standard texts on quantum mechanics. Some texts miss the fact that the scattering states are significant contributors to the perturbative energy correction. The inclusion of scattering states has wider applicability than to just the Stark effect. An explicit calculation involving a finite-square well with a perturbation is used to illustrate the importance of including scattering states into the calculation. The second-order correction to the ground-state energy is obtained in three distinct ways. The first involves altering the problem by imposing additional boundary conditions at large distances to make the positive-energy spectrum discrete. The second makes use of the continuum scattering states directly. The third bypasses the use of scattering states by solving a differential equation for the first-order correction to the wave function.