Radial Schrodinger equation

In summary, separation of variables has been used to solve the full r,0,( ) Schrodinger equation for one electron moving about a nucleus of charge Z. The 0 and (j) solutions are the spherical harmonics YL,m (0,(1>)- The bound-state radial solutions... 

Rewriting the radial Schrodinger equation with the substitution R = r F gives ... 

Solution of the Schrodinger equation for R i r), known as the radial wave functions since they are functions only of r, follows a well-known mathematical procedure to produce the solutions known as the associated Laguerre functions, of which a few are given in Table 1.2. The radius of the Bohr orbit for n = 1 is given by... 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Using the variational Cl energies and correcting these for higher order excitations, the radial Schrodinger equation solutions for E(ZP) are equal to 712.1cm and 667.4cm respectively with rmin=1.417lAand Tmax=2.93A. 

Pure rotational transitions, vibrorotational transitions and spontaneous radiative lifetimes have been derived by solving numerically [20] the one-dimensional radial part of the Schrodinger equation for the single X state preceded by construeting an interpolation... 

The bound-state energies and eigenfunctions can be obtained by solving the Schrodinger equation with boundary conditions that the radial wave function vanishes at both ends... 

The expressions of the Sections 1.5 and 1.6 are general and apply to any solution of the Schrodinger equation. In the special case of a Morse potential, the radial integrals in Eq. (1.34) can be evaluated, with some approximations, in closed form. The approximation consists in replacing the lower limit of integration by -oo. This approximation is similar to that used in Section 1.3 when obtaining the wave functions. Thus... 

In order to compute the polarizability for the lowest rovibrational level of the EF state we have to solve the radial nuclear Schrodinger equation. 

R. J. Le Roy, 2002, LEVEL 7.5 a Computer Program for Solving the Radial Schrodinger Equations for Bound and Quasibound Levels, University of Waterloo Chemical Physics Research Report CP-655. 

We have vibrationally averaged the CAS /daug-cc-pVQZ dipole and quadmpole polarizability tensor radial functions (equation (14)) with two different sets of vibrational wavefunctions j(i )). One was obtained by solving the one-dimensional Schrodinger equation for nuclear motion (equation (16)) with the CAS /daug-cc-pVQZ PEC and the other with an experimental RKR curve [70]. Both potentials provide identical vibrational... 

In general, the Slater function is not an exact solution of any Schrodinger equation (except the Is- wavefunction, which is the exact solution for the hydrogen-atom problem). Nevertheless, asymptotically, the orbital exponent C is directly related to the energy eigenvalue of that state. Actually, at large distances from the center of the atom, the potential is zero. Schrodinger s equation for the radial function R(r) is... 

The radial part of the Schrodinger equation in spherical coordinates is... 

A little more complicated system is the de-excitation of He(2 P) by Ne, where the deexcitation is dominated by the excitation transfer and only a minor contribution from the Penning ionization is involved. The experimental cross section obtained by the pulse radiolysis method, together with the numerical calculation for the coupled-channel radial Schrodinger equation, has clearly provided the major contribution of the following excitation transfer processes to the absolute de-excitation cross sections [151] (Fig. 15) ... 

The Schrodinger equation for a single particle of mass (I moving in a central potential (one that depends only on the radial coordinate r) can be written as... 

There are both bound and continuum solutions to the radial Schrodinger equation for the attractive coulomb potential because, at energies below the asymptote the potential confines the particle between r=0 and an outer turning point, whereas at energies above the asymptote, the particle is no longer confined by an outer turning point (see the figure below). 

The radial motion of a diatomic molecule in its lowest (J=0) rotational level can be described by the following Schrodinger equation ... 

In Chapter 1 and Appendix A, the angular and radial parts of the Schrodinger equation for an electron moving in the potential of a nucleus of charge Z were obtained. 

The last equation is formally identical to the radial Schrodinger equation with a non-integer value of the angular momentum quantum number. Its spectrum is bounded from below and the discrete eigenvalues are given by... 

The radial functions in Eq.(ll) are numerical solutions of monodimensional Schrodinger equations, in which the potential corresponds to the three-dimensional one in which all degrees of freedom are frozen at their equilibrium values except that radial coordinate under consideration[40, 31]. 

The Schrodinger equation can be solved exactly for the case of the hydrogen atom (see, for example, Chapter 12 of Gasiorowicz (1974)). If distances are measured in atomic units, then the first few radial functions take the form... 

Since for sodium Rc 1.7 au, eqn (5.64) predicts that has a minimum at about 2.9 au. This is in good agreement with the curve in Fig. 5.14 that was obtained by solving the radial Schrodinger equation subject to the boundary condition eqn (5.58). 

It should be emphasized that in metals the d-states, for which tight-binding functions may be used, lie above the zero of the muffin-tin potential The reason why the tight-binding method can still be used is the following. The radial part of the Schrodinger equation is... 

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