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Photodetachment of negative ions has attracted immense interest owing to its fundamental nature and practical implications with regard to technology. In this study, we explore the quantum dynamics of the photodetachment cross section of negative ion of hydrogen H− in the perturbed one dimensional linear harmonic potential via static electric field. To this end, the quantum formula for total photodetachment cross section of the H− ion is derived by calculating the dipole matrix element in spherical coordinates. In order to obtain the detached electron wave function, we have solved the time-independent Schrödinger wave equation for the perturbed Hamiltonian of the harmonic oscillator in momentum representation. To acquire the corresponding normalized final state detached electron wave function in momentum space, we have employed an approach analogous to the WKB (Wenzel–Kramers–Brillouin) approximation. The resulting analytical formula of total photodetachment cross section depicts interesting oscillator structure that varies considerably with incident-photon energy, oscillator potential frequency, and electric field strength as elucidated by the numerical results. The current problem having close analogy with the Stark effect in charged harmonic oscillator may have potential implications in atomic and molecular physics and quantum optics.
Schrödinger wave equation has been a tremendous guide for the theoretical understanding of quantum dynamics of many quantum systems in various external potentials and fields. For instance, Schrödinger equation has been employed to understand the quantum dynamics of Stark[1] and Zeeman[2] effects. Furthermore, the electronic structures and physical properties of many-body quantum systems in condensed matter physics have been well understood from the corresponding stationary-state solution of Schrödinger equation for the energy values and corresponding wave function. However, exact analytical solution of Schrödinger equation can be found only for typical quantum systems such as electron in potential wells, hydrogen and hydrogen-like atoms and harmonic oscillator in various external fields. Photodetachment of hydrogen negative ion H− in external fields and/or walls is yet another worth mentioning example in this regard. The photodetachment of the H− ion in external fields and potentials is the rare example of physical problems where exact solution of Schrödinger equation can be found both semiclassically and quantically. As such, the study of photodetachment of negative ions subjected to aforementioned conditions is also a novel way for the understanding of the long-standing issue of correspondence between semiclassical and quantum mechanics.
In the earlier experiment of photodetachment of H− in static electric field, Bryant et al.[3] observed interesting oscillations in the total cross section that resulted in immense experimental and theoretical development on this direction. Rau and Wong[4] explained quantum mechanically that the observed oscillations in the cross section resulted from quantum interference between the outgoing detached-electron waves emanating from the source and those reflected from the field-potential barrier. Subsequently, Du and Delos[5] studied the photodetachment of H− in static electric field in momentum representation by solving the time-independent Schrödinger equation for the detached electron wave function. In order to further explain the origin and physical picture of the interesting scenario of the field-induced oscillations in the H− ion photodetachment cross section, crucial development was accomplished earlier by Du and Delos[6–8] on establishing a new semi-classical approach called closed orbit theory (COT) based on reasonable approximations. According to the prediction of COT, the oscillations in cross section arise from the quantum interference between the returning detached electron waves from the walls and/or fields and the outgoing detached electron waves. Since then, the photodetachment of H− in various fields and/or surfaces has been studied extensively.[9–16] The experimental footprints of oscillations in the photodetachment of negative ions in electric field have been observed by the photodetachment microscopy.[17–20] Photodetachment microscopy offers a wonderful practical implementation of the photodetachment process in external fields and proved to be an innovative method for the measurement of electron affinity of atoms and direct observation of the spatial structure of detached electron wave function. To further explore the efficacy of the semiclassical approach, results of COT for some physical systems have been compared with that of quantum approach.[11,13,16,21,22] In contrast to the semiclassical approach of photodetachment which employs certain approximations, however, quantum mechanics approach provides more comprehensive results as it treats the problems at the fundamental level.
In the past, the studies on photodetachment cross section of H− have been mostly accomplished in various fields and/or walls instead of potential wells and barriers. On the other hand, study of quantum systems in various potentials paved a way for the development of quantum mechanics and understanding of longstanding quantum phenomenon such as quantum tunneling, classical–quantum correspondence, etc. However, despite the significant role of potential-well problems in the understanding of many phenomena of fundamental nature and practical implications in atomic physics, nuclear physics, molecular physics, and quantum physics, photodetachment of H− near potentials is rarely studied, with few recent exceptions.[22–26] A quantum-semiclassical comparative study of photodetachment of H− in a one-dimensional linear harmonic potential has been recently presented by Zhao et al.,[22] both in position and momentum coordinates. The authors employed standard semiclassical COT and frame-transformation-theory of quantum mechanics for the study. More recently,[26] Wang and Wang investigated the photodetachment dynamics of H− ion in a harmonic potential perturbed by a time-dependent electric field based on COT formalism.
In this paper, we extend the problem of the photodetachment H− in harmonic oscillator potential (HOP)[22] and attempt to explore the impact of static electric field on the photodetachment cross section in HOP. In particular, we are interested to investigate how electric field strength and harmonic oscillator frequency affect the behavior of photodetachment cross section. This is among many of the rarest physical problems where exact photodetachment cross section of H− can be obtained. The peculiarity of HOP making it a popular potential model is the fact that any general potential can be conveniently written in the form of HOP to be approximated near the equilibrium point. Besides the aforementioned points of concern, the motivation of this study also derives from the close analogy of the current problem to the Stark effect for a charged harmonic oscillator in a uniform electric field.[27] The problem has interesting implications in quantum physics and chemistry, atomic and molecular physics, and quantum optics. In addition, HOP problems have also connection regarding the measurement of the relative photoionization cross section of Rb atoms in electric fields.[28] Furthermore, the powerful predictions of the HOP model about field-dependent spacing can be effectively exploited in measuring the energy spectrum of ammonia in strong electric field.[29]
For the calculation of photodetachment cross section of H− in HOP perturbed by the static electric field, first we will solve the time-independent Schrödinger equation for the detached electron wave function in momentum representation. Then dipole matrix element will be calculated whose modulus squared is directly proportional to quantum cross section. The impact of electric field strength, laser photon energy, and HOP frequency on the structure of photodetachment cross section will be explored with numerical results.
The layout of paper is as follows. In section
We have used atomic system of units (a.u.) unless specified otherwise.
The proposed model of photodetachment in linear harmonic oscillator potential subjected to electric field is shown schematically in Fig.
The interaction Hamiltonian of the H− ion in harmonic oscillator potential subjected to static electric field governing the motion of the detached electron in cylindrical coordinates (
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The method we have employed in the calculation of quantum photodetachment cross section of the current system can be found in,[22] and references therein. The transition of the detached electron from the initial bound state of the H− ion to the final continuum state is given by the quantum photodetachment cross section which is directly proportional to the square of dipole matrix element,[7]
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![](cpb_28_2_023201/cpb_28_2_023201_ieqn7.gif)
As usual, the bound-sate initial wave function of the active electron of the H− ion in the position coordinates is given by
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![](cpb_28_2_023201/cpb_28_2_023201_ieqn8.gif)
![](cpb_28_2_023201/cpb_28_2_023201_ieqn9.gif)
In order to compute the photodetachment cross section employing Eq. (
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![](cpb_28_2_023201/cpb_28_2_023201_ieqn11.gif)
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![](cpb_28_2_023201/cpb_28_2_023201_ieqn12.gif)
The total final-state energy of the electron bound in harmonic oscillator potential in the presence of electric field has been lowered and is given by
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![](cpb_28_2_023201/cpb_28_2_023201_ieqn13.gif)
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![](cpb_28_2_023201/cpb_28_2_023201_ieqn14.gif)
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![](cpb_28_2_023201/cpb_28_2_023201_ieqn15.gif)
Substituting the values of
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After substituting the above calculated value of modulus squared of dipole matrix element into Eq. (
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![](cpb_28_2_023201/cpb_28_2_023201_ieqn18.gif)
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Now we elaborate numerically the impact of electric-field strength, harmonic potential frequency, and incident photon energy on theoretical photodetachment cross section obtained earlier with formula given in Eq. (
In Fig.
Figure
In Fig.
The interesting modification of cross section with oscillator frequency and field strength in the perturbed harmonic oscillator linear potential while keeping other spectral parameters fixed is observed first time in this study, which needs further investigation by closed orbit theory to conform the obtained results.
In conclusion, in this study we extended the problem of quantum photodetachment cross section of hydrogen negative ion confined in a one-dimensional harmonic potential to the case when it is perturbed by static electric field. The impact of photon energy, strength of electric field, and angular frequency of oscillator potential on photodetachment has been explored in detail with numerical results. In order to obtain the final-state wave function of the detached electron, we first solved the time-independent linear Schrödinger equation for the perturbed harmonic oscillator via static electric field in momentum representation. To acquire the normalized final-state wave function in momentum coordinates, we employed an approach analogous to the WKB approximation. The calculated photodetachment cross section is endowed with irregular oscillatory pattern that varies considerably with incident photon energy, the strength of static electric field and frequency of harmonic potential. A considerable modification in the amplitude and oscillatory behavior of cross section has been observed by changing the photon energy, oscillator frequency and electric field strength while keeping other spectral parameters fixed. Because of fundamental nature of HOP and its practical implications in quantum physics, quantum chemistry, atomic and molecular physics, and quantum optics, our results might be crucial for many applications in the above mentioned areas and might motivate further research on this direction. For instance, the findings might have practical implications towards understanding the complex quantum systems in the presence of potentials and electromagnetic fields with possible applications in cavity dynamics. To explore further the expected classical–quantum correspondence of the present system, the semiclassical closed orbit theory can be exploited to determine the photodetachment cross section in future.
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