Coupled Schroedinger Equations Research Articles

Symmetry approach for a three coupled Schroedinger equation system

We consider a system of three coupled cubic Schroedinger equations arising in some physical and biomathematical problems. By using the Lie symmetry methods we get new classes of exact solutions.

Controlling Ultracold Rydberg Atoms in the Quantum Regime

We discuss the properties of Rydberg atoms in a magnetic Ioffe-Pritchard trap being commonly used in ultracold atomic physics experiments. The Hamiltonian is derived, and it is demonstrated how tight traps alter the coupling of the atom to the magnetic field. We solve the underlying Schrödinger equation of the system within a given n manifold and show that for a sufficiently large Ioffe field strength the 2n;{2}-dimensional system of coupled Schrödinger equations decays into several decoupled multicomponent equations governing the center of mass motion. An analysis of the fully quantized center of mass and electronic states is undertaken. In particular, we discuss the situation of tight center of mass confinement outlining the procedure to generate a low-dimensional ultracold Rydberg gas.

Exact solution of coupled Schrödinger equations in a stationaryn-state exponential model

The $n$-state exponential model refers to coupled Schr\"odinger equations with potential ${V}_{ij}(x)={U}_{i}{\ensuremath{\delta}}_{ij}+{V}_{ij}{e}^{\ensuremath{-}\ensuremath{\alpha}x}$. Exact solutions in terms of Meijer's $G$ functions are obtained, provided the ${V}_{ij}$ factorize according to ${V}_{ij}=\ifmmode\pm\else\textpm\fi{}{V}_{i}{V}_{j}$. The nonadiabatic transition matrix $N$ is determined.

Theoretical calculations of excited rovibrational levels of HD. Term values and transition probabilities of VUV electronic bands

In this paper, we derive the theoretical properties of rovibrational levels belonging to excited B, C, B' , and D electronic states of HD. We compute the eigenvalues and eigenfunctions of the nuclear coupled Schroedinger equations using ab initio electronic molecular properties available in the literature. Transition wavenumbers and spontaneous emission probabilities are calculated for all transitions belonging to B-X, C-X, B' -X, and D-X electronic band systems of HD when the upper rotational quantum number is below or equal to 10. We compare our results with available experimental values: the accuracy in the wavenumbers is on the order of 3 reciprocal centimetres, whereas the intensity properties are satisfactorily reproduced. The origin of the remaining discrepancies is analyzed.

Recovering the M-channel Sturm-Liouville operator from M+1 spectra

For a system of M coupled Schroedinger equations, the relationship is found between the vector-valued norming constants and M+1 spectra corresponding to the same potential matrix but different boundary conditions. Under a special choice of particular boundary conditions, this equation for norming vectors has a unique solution. The double set of norming vectors and associated spectrum of one of the M+1 boundary value problems uniquely specifies the matrix of potentials in the multichannel Schroedinger equation.

Spectra and strong decays ofcc¯andbb¯states

An outline is given for the description of heavy-quark--antiquark systems by means of a set of coupled Schroedinger equations in which permanently confined two-fermion channels are coupled to free two-boson channels. Only two-particle decays satisfying the Okubo-Zweig-Iizuka rule are taken into account. Within this context the equations make a simple determination of the spectra and decay parameters possible. A comparison with the data is presented.

ApproximateS-matrix elements for three-state curve-crossing problem

Approximate solutions are developed and tested for multichannel uniform WKB theory of inelastic scattering. Although the approximate solutions are obtained specifically for a three-channel problem, they are cast in forms suggestive of an easy generalization to problems of a larger number of channels. It is found that the modified Magnus approximation, which was used for the two-channel version, is again sufficiently accurate. The stationary phase method developed elsewhere by us is found again very efficient for evaluating integrals involved in the modified Magnus approximation and saves computing time significantly in comparison with a numerical solution method of the corresponding three-channel coupled Schroedinger equations. The results of the approximate solutions obtained are compared with those by numerical solutions of the corresponding Schroedinger equations and the equations of motion in uniform WKB theory. The agreements are found excellent.

Validity of the dwba for inelastic scattering from nuclei

Coupling between ground and excited states was suggested to limit the usefulness of the distorted-wave Born approximation (DWBA) to a rather narrow limit of deformations. Comparison of calculations for the scattering of 10 to 22- Mev protons by Fe/sup 56/ as made with the DWBA and with coupled Schroedinger equations show that the DWBA has a considerably wider range of validity. (D.C.W.)

Electron Waves in the Magnetic Dipole Field of a Neutron

The relativistic wave functions for an electron moving in the field of a central magnetic dipole that has spin $\frac{1}{2}\ensuremath{\hbar}$ are studied. Spin inertia and retardation of the central field are neglected. In general, there are then four, first-order, coupled differential equations for the radial dependence of the electron waves; and these are equivalent to two coupled Schroedinger equations having a positive definite potential energy operator. In the particular case that the total angular momentum of the system is zero, however, the equations are equivalent to a single Schroedinger equation in which the potential energy is positive at all radii only for the electron-neutron interaction. The positron-neutron potential becomes strongly negative in a small region near a separation of the order of $\frac{{e}^{2}}{M{c}^{2}}$, where $M$ is the mass of the neutron. The attraction is just sufficient to form a bound ${P}_{\frac{1}{2}}$-state of the positron with zero binding energy. The reality of such neutron-positron state is open to question because of the simplifications that have been made and also because the magnetic dipole moment of the neutron is most probably of extended origin. The single equation obtained when the total angular momentum vanishes is the Mathieu differential equation which may be solved in terms of an integral function by a method developed by Lindemann. Asymptotic forms of these solutions, as well as of the fourth-order equation for total angular momentum unity are presented. The results indicate agreement with those obtained in various applications of perturbation methods. In particular, the ambiguity of the wave functions discussed previously by Bloch is apparent in the solutions for unit angular momentum.