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Journal of Physics B Atomic Molecular and Optical Physics

Coherent electron beam excitation of a model hydrogen atom

M. S. Pindzola, M. C. Witthoeft, F. Robicheaux

Published Oct 23, 2000• 5 citations
Publisher

Abstract

Excitation of a model hydrogen atom by a coherent electron beam is calculated by direct solution of the time-dependent Schrödinger equation on a three-dimensional lattice. The Coulomb interaction between the electrons is described by v ( r 1 , r 2 ) = 1/r > and all angular momenta are set to zero. The coherent beam consists of moving wavepackets for each of two free electrons which arrive at the third bound electron at different times. As recently predicted by Robicheaux and Noordam (2000 Phys. Rev. Lett. 84 3735), the probability for a weak excitation should vary as a function of arrival time. The time-dependent calculations for the model hydrogen atom show that the 1s → 2s excitation probability oscillates about an axis which is between one and two times the excitation probability from only one wavepacket. The frequency of the probability oscillation is found to be equal to the 1s → 2s transition frequency. Large destructive interference of the 1s → 2s cross section is illustrated by contour maps of the 2s projected probability as a function of wavepacket arrival time.

DOI
10.1088/0953-4075/33/21/107

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Excitation of a model hydrogen atom by a coherent electron beam is calculated by direct solution of the time-dependent Schrödinger equation on a three-dimensional lattice. The Coulomb interaction between the electrons is described by v ( r 1 , r 2 ) = 1/r > and all angular momenta are set to zero. The coherent beam consists of moving wavepackets for each of two free electrons which arrive at the third bound electron at different times. As recently predicted by Robicheaux and Noordam (2000 Phys. Rev. Lett. 84 3735), the probability for a weak excitation should vary as a function of arrival time. The time-dependent calculations for the model hydrogen atom show that the 1s → 2s excitation probability oscillates about an axis which is between one and two times the excitation probability from only one wavepacket. The frequency of the probability oscillation is found to be equal to the 1s → 2s transition frequency. Large destructive interference of the 1s → 2s cross section is illustrated by contour maps of the 2s projected probability as a function of wavepacket arrival time.
We present a fully quantum model for the excitation of a bound electron based on the “free-electron bound-electron resonant interaction” (FEBERI) scheme. The bound electron is modeled as a quantum two-level system (TLS) at any initial quantum (qubit) state, and the free electron is presented as a pre-shaped quantum electron wavepacket (QEW). In the case that the QEW is short or modulated at optical frequency, the TLS quantum state may be coherently controlled with multiple modulation-correlated QEWs. For this case, we derive the transition probability of the TLS due to interaction with a multi-particle beam based on an analytical approximate solution of the Schrodinger equation that amounts to using Born’s probabilistic interpretation of the quantum electron wavefunction. We verify the credibility of the analytical model at its validity ranges using a fully quantum density matrix computation procedure. It is shown that the transition probability can grow quadratically with the number of correlated QEWs and exhibit Rabi oscillation. The study indicates a possibility of engineering the quantum state of a TLS by utilizing a beam of shaped QEWs.
The interference between quantum amplitude for two electrons, emitted from two source points, to be detected at two detection points, is a direct result of quantum exchange statistics. Such interference is observed in the coincidence probability, compared with that of statistically independent electrons, by computing the time correlation function from the arrival times of the electrons. When the two detectors are separated by a distance less than the coherence length, the coincidence probability is suppressed for electrons (antibunching) due to the Pauli principle, even though they do not interact with each other. However, electrons are charged particles. The Coulomb potential, which governs the scattering of one charged particle by another, is so long ranged. It is obvious that we must consider the Pauli principle and the Coulomb interactions simultaneously. This paper deals with basic experimental and theoretical investigations of the antibunching behavior of electrons in a free beam by considering the Pauli principle and the direct Coulomb interaction between two individual electrons. The experimentally found dependences are described in a model which considers the Coulomb scattering and theoretical values of correlation signals evaluated by analytical calculations agree with those determined by experiment. A study of the time correlation function from the arrival times of the electrons will lead to an understanding of the physical processes that take place in electron guns.
This paper is a theoretical discussion of a "coherence effect" in the excitation of hydrogen atoms to the n=3n=3n=3 state by electron bombardment. By coherence is meant that the density matrix describing the atom after the collision has off-diagonal matrix elements in the field-free basis. The differential cross section for producing Balmer-$\ensuremath{\alpha}$ photons at 90\ifmmode^\circ\else\textdegree\fi{} to the electron beam is calculated as a function of an applied longitudinal electric field. The off-diagonal elements show up as a dependence of the observed light on the directions of the electric field. Theoretical results based on the Born approximation are compared with an experiment at 200 and 500 eV. It is pointed out that this method for measuring relative phases of scattering amplitudes can be generally applied and is not restricted to n=3n=3n=3 excitation of hydrogen.
The Bohr model is highly successful in explaining many features of the hydrogen atom. The ground state wave function does not have any angular dependence. The normalized hydrogen atom wave functions are directly accessible in Mathematica and can be readily verified as solutions of the Schrodinger equation. Get the hydrogen atom ground state wave function in terms of the Bohr radius a. The probability of finding the electron per volume at a given location is given by the square of the wave function. The electron has only a very tiny probability of being found inside the proton, whose size is about 1 fm.
The quantum dynamics of a hydrogen-like atom confined in one-dimensional box with oscillating walls is studied.The description of the system is reduced to a one-dimensional Schrödinger equation for Coulomb potential with time-dependent boundary conditions, which is solved numerically.Using the obtained solution, the average kinetic energy and binding energies are calculated as a function of time.It is found that both the average kinetic energy and the binding energies are periodic in time with the period depending on the wall's oscillation parameters.The probability density is also analyzed as a function of time and coordinate.
The purely algebraic technique associated with the creation and annihilation operators to resolve the radial equation of Hydrogen-like atoms (HLA) for generating the bound energy spectrum and the corresponding wave functions is suitable for many calculations in quantum physics. However, the analytical approach with series is extensively used based on wave mechanics theory in most of quantum textbooks. Indeed, much More complete than the old solution of Schrödinger's time-independent differential equation (TISE), one can simply earn all quantum information of a system by using the operational method. In addition to earlier two models, including the quantum harmonic oscillator and the total angular momentum, it can undoubtedly be a third fundamental model to solve Schrödinger's eigenvalue equation of the HLA systems in Hilbert space similar to the harmonic oscillator. We will illustrate how systematically making an appropriate groundwork to discover the coherent states can lead to providing the energy quantization and normalized radial wave functions attached to the matrix representation without additional assumptions.