The photoionization of a one-dimensional Dirac delta-function potential is examined analytically for the high-frequency range in which the individual photon energy exceeds the binding energy. Closed-form expressions are presented for the population amplitudes, the spatial wavefunction, and the probability flux with no approximations other than the rotating-wave approximations, the instantaneous turn-on approximation, and the neglect of continuum-continuum transitions. The expressions are examined in various levels of approximation and compared with what would be expected for exponential decay.
The photoionization of a one-dimensional Dirac delta-function potential is examined analytically for the high-frequency range in which the individual photon energy exceeds the binding energy. Closed-form expressions are presented for the population amplitudes, the spatial wavefunction, and the probability flux with no approximations other than the rotating-wave approximations, the instantaneous turn-on approximation, and the neglect of continuum-continuum transitions. The expressions are examined in various levels of approximation and compared with what would be expected for exponential decay.
This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.
We consider the time evolution of a particle subjected to both a uniform electrostatic field F and a one-dimensional delta-function potential well. We derive the propagator K_F(x,t|x',0) of this system, directly leading to the wavefunction psi_F(x,t), in which its essential ingredient K_F(0,t|0,0), accounting for the ionization-recombination in the bound-continuum transition, is exactly expressed in terms of the multiple hypergeometric functions F(z_1,z_2,...,z_n). And then we obtain the ingredient K_F(0,t|0,0) in an appropriate approximation scheme, expressed in terms of the generalized hypergeometric functions p_F_q(z) being much more transparent to physically interpret and much more accessible in their numerical evaluation than the functions F(z_1,z_2,...,z_n).
Abstract. 1-dimensional hamiltonian with ionization energy (ξ) is shown to be exactly the same with the total energy from the standard harmonic oscillator hamiltonian, with the harmonic oscillator potential, mω 2 x 2 /2. Commutation relations for the ladder operators, energy at the n th state and the expectation value for the potential are analyzed and shown to be consistent with the principle of ξ. We also derive 1-dimensional wave functions for Dirac delta (−αδ(x)) potential and with V (x) = 0. Formal justifications are given as to why the total energy in ξ based Fermi-Dirac statistics is exactly the same with the standard Fermi-Dirac statistics. Energy levels and Bohr radius of a Hydrogen atom are also shown to be consistent with the ionization energy concept. This concept is also applied in a free Mn atom that represent a free many-electron atom. 1.
We consider the time evolution of a particle bound by an attractive one-dimensional delta-function potential (at x = 0) when a uniform electrostatic field (F) is applied. We explore explicit expressions for the time-dependent wavefunction ψ_F(x,t) and the ionization probability {\mathcal{P}}(t), respectively, in the weak-field limit. In doing so, ψ_F(0,t) is a key element to their evaluation. We obtain a closed expression for ψ_F(0,t) which is an excellent approximation of the exact result being a numerical solution of the Lippmann-Schwinger integral equation. The resulting probability density |ψ_F(0,t)|^2, as a simple alternative to {\mathcal{P}}(t), is also in good agreement to its counterpart from the exact one. In doing this, we also find a new and useful integral identity of the Airy function.
In this work we attack analytically a class of one dimensional time-dependent potentials based on a δ(x) potential whose intensity varies on time as v(t), that is, V (x; t) = v(t) δ(x). Although this model in principle may appear too simple, a kind of toy model, it turns out to be very dfficult to handle without any kind of approximation. To make progress, apart from the local potential V (x; t) already mentioned, a type of non-local potentials are considered. To clarify the general results, various examples, with different initial conditions, are studied in detail.
The dynamics of the bound-state populations and the photoelectron spectra are calculated for model atomic systems irradiated by a pulse with a harmonically modulated frequency. For a system including a single bound state a model is adopted, which takes into account the threshold behavior typical of negative ions and admits continuum-continuum transitions. For a model one-dimensional atom the results are obtained through a numerical integration of the Schr\"odinger equation. The case of a resonant ionization is also considered. The modulation of the frequency results in changing the instantaneous rate of decay of the initial state, in inducing additional population oscillations for multilevel systems, and in additional peaks in the photoelectron spectra.