Physical Review A

First-order perturbed wave functions for the hydrogen atom in a harmonic uniform external electric field

Viorica Florescu, Tudor A. Marian

Published Dec 1, 1986• 34 citations

Abstract

We investigate the linear response of a nonrelativistic hydrogenlike atom to a harmonic uniform electric field, which is switched on adiabatically. For the electron initially in an arbitrary bound \ensuremath{\Vert}nlm〉 state, the first-order correction to the wave function has been written in terms of a vector function ${\mathrm{w}}_{\mathrm{nlm}}$. We report an exact closed-form expression of the vector ${\mathrm{w}}_{\mathrm{nlm}}$ that we have derived using the Coulomb Green's function. Our general analytic results prove to be useful in a systematic study of two-photon processes involving the hydrogen atom.

DOI: 10.1103/physreva.34.4641

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Physical Review A

First-order perturbed wave functions for the hydrogen atom in a harmonic uniform external electric field

Viorica Florescu, Tudor A. Marian

Published Dec 1, 1986• 34 citations

Publisher

Abstract

{\mathrm{w}}_{\mathrm{nlm}}

. We report an exact closed-form expression of the vector

{\mathrm{w}}_{\mathrm{nlm}}

that we have derived using the Coulomb Green's function. Our general analytic results prove to be useful in a systematic study of two-photon processes involving the hydrogen atom.

DOI: 10.1103/physreva.34.4641

Related papers

Linear response of a nonrelativistic hydrogenlike atom to a single-mode radiation field. II. Electric dipole approximation: General formalism

Tudor A. Marian

The interaction of a low-intensity laser field with a nonrelativistic one-electron atom is considered to the first order of perturbation theory and in the electric dipole approximation. The radiation field is turned on adiabatically modifying the initial unperturbed atomic state, which is either an angular-momentum eigenstate $|\mathrm{nlm}〉$ or a Stark state $|{\mathrm{nn}}_{e}m〉$ . The first-order correction to the wave function is expressed both in the length and velocity gauges in terms of a vector function called the linear-response vector and depending on the field-free energy eigenstate. We derive in a unifying manner the linear-response vectors in the position representation, as closed-form contour integrals, starting from a unique generating function built up with the Coulomb Green function. The linear-response vectors are then evaluated in momentum space via a Fourier transformation: they are obtained as integral representations and also in explicit form, as generalized hypergeometric functions. With reference to the static limit, we complement earlier results and find the reduced linear-response vectors in momentum space, as Fourier transforms of their coordinate-representation counterparts. The low-frequency behavior of the length-gauge first-order correction to the ground-state wave function is established in coordinate as well as in momentum space. We finally point out the high-frequency limit of the linear response in the velocity gauge.The interaction of a low-intensity laser field with a nonrelativistic one-electron atom is considered to the first order of perturbation theory and in the electric dipole approximation. The radiation field is turned on adiabatically modifying the initial unperturbed atomic state, which is either an angular-momentum eigenstate $|\mathrm{nlm}〉$ or a Stark state $|{\mathrm{nn}}_{e}m〉$. The first-order correction to the wave function is expressed both in the length and velocity gauges in terms of a vector function called the linear-response vector and depending on the field-free energy eigenstate. We derive in a unifying manner the linear-response vectors in the position representation, as closed-form contour integrals, starting from a unique generating function built up with the Coulomb Green function. The linear-response vectors are then evaluated in momentum space via a Fourier transformation: they are obtained as integral representations and also in explicit form, as generalized hypergeometric functions. With reference to the static limit, we complement earlier results and find the reduced linear-response vectors in momentum space, as Fourier transforms of their coordinate-representation counterparts. The low-frequency behavior of the length-gauge first-order correction to the ground-state wave function is established in coordinate as well as in momentum space. We finally point out the high-frequency limit of the linear response in the velocity gauge.

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Bound-bound two-photon transition matrix elements for the hydrogen atom in the dipole approximation

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By use of the linear-response-vectors method, the Kramers-Heisenberg matrix elements for arbitrary bound-bound transitions of a nonrelativistic hydrogenlike atom have been evaluated. To illustrate the discussion, Rayleigh and Raman scattering have been selected as relevant examples of second-order processes. We have derived, in a unitary manner, A- and \ensuremath{\varphi}-gauge compact analytic formulas, both for angular momentum and Stark states. Our results are expressed in terms of two new generalized hypergeometric functions with four variables, $_{1}\mathrm{F}_{\mathrm{E}}$ and $_{2}\mathrm{F}_{\mathrm{E}}$ , which are finite sums of Gauss hypergeometric functions. Special attention is given to the two-photon transitions involving the atomic ground state. For angular momentum states, we mention the connection that we have established between our general results and earlier ones.By use of the linear-response-vectors method, the Kramers-Heisenberg matrix elements for arbitrary bound-bound transitions of a nonrelativistic hydrogenlike atom have been evaluated. To illustrate the discussion, Rayleigh and Raman scattering have been selected as relevant examples of second-order processes. We have derived, in a unitary manner, A- and \ensuremath{\varphi}-gauge compact analytic formulas, both for angular momentum and Stark states. Our results are expressed in terms of two new generalized hypergeometric functions with four variables, $_{1}\mathrm{F}_{\mathrm{E}}$ and $_{2}\mathrm{F}_{\mathrm{E}}$, which are finite sums of Gauss hypergeometric functions. Special attention is given to the two-photon transitions involving the atomic ground state. For angular momentum states, we mention the connection that we have established between our general results and earlier ones.

Physical Review AApr 1, 198911 citations

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

Gregory S. Adkins, R F Hood

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European Journal of PhysicsJan 1, 19897 citations

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One- and two-photon transitions in the hydrogen atom are analytically evaluated in the absence and in the presence of an external electric field. The emission probabilities are different for the hydrogen and anti-hydrogen atoms due to the existence of contributions, linear in electric field. The magnitude of these contributions is evaluated within the nonrelativistic limit. The Coulomb Green function method is applied. Different nonrelativistic "forms" for the decay probabilities in combination with different gauge choices are considered. The three-photon E1E1E1 2p-1s transition probability is also evaluated and possible applications of the results are discussed.One- and two-photon transitions in the hydrogen atom are analytically evaluated in the absence and in the presence of an external electric field. The emission probabilities are different for the hydrogen and anti-hydrogen atoms due to the existence of contributions, linear in electric field. The magnitude of these contributions is evaluated within the nonrelativistic limit. The Coulomb Green function method is applied. Different nonrelativistic "forms" for the decay probabilities in combination with different gauge choices are considered. The three-photon E1E1E1 2p-1s transition probability is also evaluated and possible applications of the results are discussed.

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Field dependence of the hydrogen Stark line intensities

Aleksandr A. Kamenski, Vitalii D Ovsiannikov

General equations are presented for determining the field-dependent variation of the wavefunctions, matrix elements and intensities of dipole transitions between Stark levels of a hydrogen atom in a field range where the fine-structure effects may be neglected. The Sturm-series resolution for the reduced Coulomb-Green function is used to derive analytical expressions for the first-order corrections in terms of parabolic quantum numbers of initial and final states. Simple forms of these expressions as functions of the upper-state quantum numbers are presented explicitly for the Lyman and Balmer series. The selective action of the dc electric field on the intensity of Stark components of hydrogen lines is revealed in numerical calculations for both and types of emitted radiation. This selectivity may cause, in particular, enhancement of the high-frequency and damping of the low-frequency Stark -lines of the Lyman series.General equations are presented for determining the field-dependent variation of the wavefunctions, matrix elements and intensities of dipole transitions between Stark levels of a hydrogen atom in a field range where the fine-structure effects may be neglected. The Sturm-series resolution for the reduced Coulomb-Green function is used to derive analytical expressions for the first-order corrections in terms of parabolic quantum numbers of initial and final states. Simple forms of these expressions as functions of the upper-state quantum numbers are presented explicitly for the Lyman and Balmer series. The selective action of the dc electric field on the intensity of Stark components of hydrogen lines is revealed in numerical calculations for both and types of emitted radiation. This selectivity may cause, in particular, enhancement of the high-frequency and damping of the low-frequency Stark -lines of the Lyman series.

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Radiative decay of hydrogenlike atoms in an electric field

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Photon emission by an atom in the metastable $2S$ state in a weak static electric field is examined within the context of bound-state quantum electrodynamics. Low's method is applied to obtain the expansion in terms of Coulomb wave functions of the propagation function for an electron interacting with the nuclear Coulomb field, the applied electric field, and the quantized radiation field. For a weak applied field, and to lowest relative order in $\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}$, the $2S$ -state resonant contribution to the propagation function can be factorized to provide the basis for a calculation of the angular distribution of single-photon radiation and lifetime of the excited state. A numerical evaluation is given, which takes into account effects of electric dipole, magnetic dipole, and magnetic quadrupole emission, relativistic corrections, and finite level widths.Photon emission by an atom in the metastable $2S$ state in a weak static electric field is examined within the context of bound-state quantum electrodynamics. Low's method is applied to obtain the expansion in terms of Coulomb wave functions of the propagation function for an electron interacting with the nuclear Coulomb field, the applied electric field, and the quantized radiation field. For a weak applied field, and to lowest relative order in $\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}$, the $2S$-state resonant contribution to the propagation function can be factorized to provide the basis for a calculation of the angular distribution of single-photon radiation and lifetime of the excited state. A numerical evaluation is given, which takes into account effects of electric dipole, magnetic dipole, and magnetic quadrupole emission, relativistic corrections, and finite level widths.

Physical Review AJan 1, 198050 citations

{\mathrm{w}}_{\mathrm{nlm}}

. We report an exact closed-form expression of the vector

{\mathrm{w}}_{\mathrm{nlm}}

that we have derived using the Coulomb Green's function. Our general analytic results prove to be useful in a systematic study of two-photon processes involving the hydrogen atom.

|\mathrm{nlm}〉

or a Stark state

|{\mathrm{nn}}_{e}m〉

. The first-order correction to the wave function is expressed both in the length and velocity gauges in terms of a vector function called the linear-response vector and depending on the field-free energy eigenstate. We derive in a unifying manner the linear-response vectors in the position representation, as closed-form contour integrals, starting from a unique generating function built up with the Coulomb Green function. The linear-response vectors are then evaluated in momentum space via a Fourier transformation: they are obtained as integral representations and also in explicit form, as generalized hypergeometric functions. With reference to the static limit, we complement earlier results and find the reduced linear-response vectors in momentum space, as Fourier transforms of their coordinate-representation counterparts. The low-frequency behavior of the length-gauge first-order correction to the ground-state wave function is established in coordinate as well as in momentum space. We finally point out the high-frequency limit of the linear response in the velocity gauge.

_{1}\mathrm{F}_{\mathrm{E}}

and

_{2}\mathrm{F}_{\mathrm{E}}

, which are finite sums of Gauss hypergeometric functions. Special attention is given to the two-photon transitions involving the atomic ground state. For angular momentum states, we mention the connection that we have established between our general results and earlier ones.

Photon emission by an atom in the metastable

2S

state in a weak static electric field is examined within the context of bound-state quantum electrodynamics. Low's method is applied to obtain the expansion in terms of Coulomb wave functions of the propagation function for an electron interacting with the nuclear Coulomb field, the applied electric field, and the quantized radiation field. For a weak applied field, and to lowest relative order in $\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}$, the

2S

-state resonant contribution to the propagation function can be factorized to provide the basis for a calculation of the angular distribution of single-photon radiation and lifetime of the excited state. A numerical evaluation is given, which takes into account effects of electric dipole, magnetic dipole, and magnetic quadrupole emission, relativistic corrections, and finite level widths.