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arXiv:2111.06984 · Physical Review A

Exact energy eigenstates of the Coulomb-Stark Hamiltonian

Seyedmohammad Yusofsani, M. Kolesík

Published Mar 3, 2022• 0 citations
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Abstract

An approximation-free, numerically efficient algorithm is presented for the Hamiltonian eigenstates of the Stark hydrogen problem describing a quantum particle exposed to the central Coulomb force and a homogeneous external field. As an example of application in a state expansion with continuous energy, we calculate the time-dependent wave function of an electron tunneling from a hydrogen atom suddenly exposed to an external electric field.

DOI
10.1103/physreva.105.032203

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An approximation-free, numerically efficient algorithm is presented for the Hamiltonian eigenstates of the Stark hydrogen problem describing a quantum particle exposed to the central Coulomb force and a homogeneous external field. As an example of application in a state expansion with continuous energy, we calculate the time-dependent wave function of an electron tunneling from a hydrogen atom suddenly exposed to an external electric field.
Nondispersive wave packets in a fictitious time variable are calculated analytically for the field-free hydrogen atom. As is well known by means of the Kustaanheimo-Stiefel transformation the Coulomb problem can be converted into that of a four-dimensional harmonic oscillator, subject to a constraint. This regularization makes use of a fictitious time variable, but arbitrary Gaussian wave packets in that time variable in general violate that constraint. The set of ``restricted Gaussian wave packets'' consistent with the constraint is constructed and shown to provide a complete basis for the expansion of states in the original three-dimensional coordinate space. Using that expansion arbitrary localized Gaussian wave packets of the hydrogen atom can be propagated analytically and exhibit a nondispersive periodic behavior as functions of the fictitious time. Restricted wave packets with and without well-defined angular momentum quantum numbers are constructed. They will be used as trial functions in time-dependent variational computations for the hydrogen atom in static external fields in the subsequent paper [T. Fab\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \check{c}\else \v{c}\fi{}, J. Main, and G. Wunner, Phys. Rev. A 79, 043417 (2009)].
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 time-evolution of a wave packet in a perfectly periodic lattice potential is considered for an electron that is in a zero-field Bloch energy eigenstate prior to the adiabatic application of a uniform d.c. electric field. Power series expansions in time are constructed for both the exact wave function and the Houston wave function. A comparison of the coefficients in these two expansions shows that the Houston function diverges instantly and drastically from the exact wave function. A special-case example is presented for which the exact wave function is pure-real for all values of time whereas the corresponding Houston function is not.
The problem of the motion of a particle attracted by two fixed centers of force according to the Coulomb force law can be treated by classical mechanics and has been used in quantum theory by Pauli and Messen for a theory of the hydrogen molecule ion.1 In the quantum mechanics, where the energy levels are determined as the “eigenwerte” of Schrodinger’s equation, the variables are separable and the boundary value problem is easily set up. But thus far a satisfactory treatment of the differential equations involved is lacking. Burrau2 has recently carried out a numerical integration of the problem for the lowest energy level of an electron moving under the influence of two fixed centers of Coulomb attraction as a function of the distance apart of these centers. In this paper, Burrau’s data are used to give a semi-quantitative discussion of the neutral hydrogen molecule. His values are: Nuclear separation 1.0 1.3 1.6 1.8 2.0 2.2 2.4 2.95 Electronic energy 2.896 2.648 2.436 2.309 2.204 2.109 2.025 1.836 The unit of separation is the Bohr l1 orbit radius of hydrogen atom, that of energy is the ionization potential of atomic hydrogen.
The hydrogen atom with the Coulomb interaction is one of the exactly solvable non-relativistic quantum models. Unlike many other exactly solvable models it describes a real physical object providing the formulas for energy levels and stationary state wave functions of a real hydrogen atom. In this paper we modify the model replacing the Coulomb interaction by the interaction of the proton and the electron with the classical electromagnetic field serving as an intermediary transmitting the electromagnetic interaction of these two charged quantum particles.