A Symbolic-Numerical Algorithm for Solving the Eigenvalue Problem for a Hydrogen Atom in the Magnetic Field: Cylindrical Coordinates (original) (raw)
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We describe POTHMF, a program to compute matrix elements of the coupled radial equations for a hydrogen-like atom in a homogeneous magnetic field. POTHMF computes with a prescribed accuracy the oblate angular spheroidal functions, which depend on a parameter and corresponding eigenvalues, and the matrix elements, which are integrals of the eigenfunctions multiplied by their derivatives with respect to the parameter. The program, implemented in Maple-Fortran, consists of a package of symbolic-numerical algorithms that reduce a singular two-dimensional boundary value problem for an elliptic second-order partial differential equation to a regular boundary value problem for a system of second-order ordinary differential equations using the Kantorovich method.
Chemical Physics, 2018
The mapping of an electronic state on a real-space support lattice may offer advantages over a basis set ansatz in cases where there are linear dependences due to basis set overcompleteness or when strong internal or external fields are present. Such discretization methods are also of interest because they allow for the convenient numerical integration of matrix elements of local operators. We have developed a pseudo-spectral approach to the numerical solution of the time-dependent and time-independent Schrödinger equations describing electronic motion in atoms and atomic ions in terms of a spherical coordinate system. A key feature of this scheme is the construction of a Variational Basis Representation (VBR) for the non-local component and of a Generalized Finite Basis Representation (GFBR) for the local component of the electronic Hamiltonian operator. Radial Hamiltonian eigenfunctions r () nl; of the H atom-like system and spherical harmonics form the basis set. Two special cases of this approach are explored: In one case, the functions of the field-free H atom are used as the elements of the basis set, and in the second case, each radial basis function has been obtained by variationally optimizing a shielding parameter β to yield a minimum energy for a particular eigenstate of the H atom in a uniform magnetic field. We derive a new quadrature rule of nearly Gaussian accuracy for the computation of matrix elements of local operators between radial basis functions and perform numerical evaluation of local operator matrix elements involving spherical harmonics. With this combination of radial and angular quadrature prescriptions we ensure to a good approximation the discrete orthogonality of Hamiltonian eigenfunctions of H atom-like systems for summation over the grid points. We further show that sets of r () nl; functions are linearly independent, whereas sets of the polar-angle-dependent components of the spherical harmonics, i.e., the associated Legendre functions, are not and provide a physical interpretation of this mathematical observation. The pseudo-spectral approach presented here is applied to two model systems: the field-free H atom and the H atom in a uniform magnetic field. The results demonstrate the potential of this method for the description of challenging systems such as highly charged atomic ions.
Computer Physics Communications, 2008
A FORTRAN 77 program is presented which calculates with the relative machine precision potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. The potential curves are eigenvalues corresponding to the angular oblate spheroidal functions that compose adiabatic basis which depends on the radial variable as a parameter. The matrix elements of radial coupling are integrals in angular variables of the following two types: product of angular functions and the first derivative of angular functions in parameter, and product of the first derivatives of angular functions in parameter, respectively. The program calculates also the angular part of the dipole transition matrix elements (in the length form) expressed as integrals in angular variables involving product of a dipole operator and angular functions. Moreover, the program calculates asymptotic regular and irregular matrix solutions of the coupled adiabatic radial equations at the end of interval in radial variable needed for solving a multi-channel scattering problem by the generalized R-matrix method. Potential curves and radial matrix elements computed by the POTHMF program can be used for solving the bound state and multichannel scattering problems. As a test desk, the program is applied to the calculation of the energy values, a short-range reaction matrix and corresponding wave functions with the help of the KANTBP program. Benchmark calculations for the known photoionization cross-sections are presented.
Simple method for numerical solving of Schroedinger equation for hydrogen atom in electric field
Nuclear Technology and Radiation Protection
A prop a ga tion nu mer i cal method for de ter min ing en ergy eigenvalues and eigen wave functions for hy dro gen atom in con stant and uni form elec tric field is de scribed in this pa per. So lution is pre sented for 3-D Schroedinger equa tion in nat u ral par a bolic co-or di nate sys tem. Crite ria for ac cept ing eigenvalues are in tro duced, and re sults are com pared with pre vi ous pa pers.
THE ENERGY EIGENVALUES OF THE TWO DIMENSIONAL HYDROGEN ATOM IN A MAGNETIC FIELD
International Journal of Modern Physics E, 2006
In this paper, the energy eigenvalues of the two dimensional hydrogen atom are presented for the arbitrary Larmor frequencies by using the asymptotic iteration method. We first show the energy eigenvalues for the no magnetic field case analytically, and then we obtain the energy eigenvalues for the strong and weak magnetic field cases within an iterative approach for n = 2 − 10 and m = 0 − 1 states for several different arbitrary Larmor frequencies. The effect of the magnetic field on the energy eigenvalues is determined precisely. The results are in excellent agreement with the findings of the other methods and our method works for the cases where the others fail.
2012
Symbolic-numeric solving of the boundary value problem for the Schrödinger equation in cylindrical coordinates is given. This problem describes the impurity states of a quantum wire or a hydrogen-like atom in a strong homogeneous magnetic field. It is solved by applying the Kantorovich method that reduces the problem to the boundaryvalue problem for a set of ordinary differential equations with respect to the longitudinal variables. The effective potentials of these equations are given by integrals over the transverse variable. The integrands are products of the transverse basis functions depending on the longitudinal variable as a parameter and their first derivatives. To solve the problem at high magnetic quantum numbers |m| and study its solutions we present an algorithm implemented in Maple that allows to obtain analytic expressions for the effective potentials and for the transverse dipole moment matrix elements. The efficiency and accuracy of the derived algorithm and that of Kantorovich numerical scheme are confirmed by calculating eigenenergies and eigenfunctions, dipole moments and decay rates of low-excited Rydberg states at high |m| ∼ 200 of a hydrogen atom in the laboratory homogeneous magnetic field γ ∼ 2.35 × 10 −5 (B ∼ 6T ).
Exact solution for a hydrogen atom in a magnetic field of arbitrary strength
An exact solution describing the quantum states of a hydrogen atom in a homogeneous magnetic field of arbitrary strength is obtained in the form of a power series in the radial variable with coefficients being polynomials in the sine of the polar angle. Energy levels and wave functions for the ground state and for several excited states are calculated exactly for the magnetic field varying in the range 0ϽB/(m 2 e 3 c/ប 3 )р4000.
Physical review. A, 1988
Exact integral expressions and simple analytical estimates for the bound-bound, boundcontinuum, and continuum-continuum dipole matrix elements are derived by use of the momentum-space eigenfunctions for a one-dimensional model of a hydrogen atom. These results provide the essential ingredients for the numerical study of the quantum mechanisms responsible for the chaotic ionization of highly excited hydrogen atoms in intense microwave fields. A specific numerical algorithm for solving the Schrodinger equation for a one-dimensional hydrogen atom in an oscillating electric field is described which uses these results for the dipole matrix elements along with a discrete representation of the continuum. In addition, the momentum-space representation of the Sturmian basis functions has been used to derive exact integral expressions and convenient analytical estimates for the projections of the Sturmian basis functions onto the hydrogenic bound and continuum states. These results are used to provide a direct comparison of numerical calculations for the ionization of one-dimensional hydrogen atoms using the hydrogenic and Sturmian bases.
An operator method for solving the hydrogen atom
2006
We discuss an operator solution for the non-relativistic hydrogen atom. The method deals with the radial Schrödinger equation and adds a phase and its corresponding operator to the basic variables of the problem. The radial problem is then expressed in terms of those operators which, furthermore, are found to span an su(1, 1) Lie algebra. From such operators the bound energy spectrum and its set of radial eigenfunctions are computed. The approach is rather similar to the one employed for computing the angular momentum spectrum and eigenfunctions but with operators satisfying an su(1, 1) algebra instead of an su(2) one.