Numerical Solution of the Radial Schrodinger Equation

The Schrodinger equation for the hydrogen atom in a magnetic field is 

In Chapter 5 we found that in quantum mechanics L lies on the surface of a cone. A classical-m.echa.mc treatment of the motion of L in an applied magnetic field shows that the field exerts a torque on m, causing L to revolve about the direction of B at a constant frequency given by 15 (Itt L, while maintaining a constant angle with B. This 

Recall from the discussion after (6.81) that if R r) behaves as 1/r near the origin, then if 1, F(r) is not quadratically integrable also, the value ft = 1 is not allowed, as noted after (6.83). Hence F(r) = rR r) must be zero at r = 0. 

For I 0, G(r) in (6.138) is infinite at r = 0, which upsets most computers. To avoid this problem, one starts the solution at an extremely small value of r (for example, 10 for the dimensionless r ) and approximates F r) as zero at this point. 

As an example, we shall use the Numerov method to solve for the lowest bound-state H-atom energies. Here, V = -c /dtrsor = —e jr, where e = e/ AttbqYI. The radial equation (6.62) contains the three constants and h, where e = ej(4Treo) has SI units of m (see Table A.l of the Appendix) and hence has the dimensions [e ] = Following the procedure used to derive Eq. (4.73), we find the 

For a one-particle central-force problem, the wave function is given by (6.16) as (/f = R r)Yf 0, l ) and the radial factor R(r) is found by solving the radial equation 

The bound-state H-atom energies are all less than zero. Suppose we want to find the H-atom bound-state eigenvalues with E, —0.04. Equating this energy to V, we have (Problem 6.40) -0.04 = -1/r,. and the classically allowed region for this energy value extends from = 0 to = 25. Going two units into the classically forbidden 

Gr in (6.140) conrains the parameter /, so the program of Table 4.1 has to be modified to input the value of L When setting up a spr dsheet, enter the / value in some cell and refer to this c ell when you type the formula for cell B7 (Rg. 4.8) that defines G,. Start column A at W = 1 X 10. Column C of the spreadsheet will contain F, values instead of tj/r valu and F will differ negligibly from zero at r,. = 1 x 10 and will be taken as zero at this point. 

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The complex phase shift can be obtained from exact numerical solution of the radial Schrodinger equation.2 The following quantities can immediately be given in terms of 8r The differential elastic cross section in the center-of-mass system... 

These results were then numerically checked with those of Vrscay (1983, 1985) by suitably replacing the values for k and n. Finally, the substitution of N = 2, n = 0, and = 0 provided the additional information for the construction of two-point Pade approximants. The results were compared with the numerical solution of the radial Schrodinger equation for the ground state, obtaining excellent agreement (Adams, 1987). 

In this part of the paper we present the study of the stabilization of the multistep exponentially-fitted methods. More specifically we present a family of singularly P-stable exponentially-fitted four-step methods and a family of six-step exponentially-fitted methods for the numerical solution of the radial Schrodinger equation. 

J. R. Cash, A. D. Raptis and T. E. Simos, A sixth order exponentially-fitted method for the numerical solution of the radial Schrodinger equation, J. Comput. Phys., 1990, 91, 413-423. 

G. Psihoyios and T. E. Simos, The numerical solution of the radial Schrodinger equation via a trigonometrically fitted family of seventh algebraic order Predictor-Corrector methods, J. Math. Chem., 2006, 40(3), 269-293. 

In 29 the authors have developed a trigonometrically-fitted multiderivative method for the numerical solution of the radial Schrodinger equation. The methods are called multiderivative since uses derivatives of order two and four. The method has the general form ... 

It is possible, of course, to simply take the numerical solution of the radial Schrodinger equation for the particular atom under question and use it as the radial part of the atomic orbital. It is also possible, and mathematically advantageous, to approximate the exponential decay of the radial part of the atomic orbital Rn (r) in the outer region. One particularly successful approach goes back to Slater s recipe [13],... 

Section 6.9 Numerical Solution Of The Radial Schrodinger Equation 157... 

In order to show the efficieney of the new methodology we will study the error analysis and we will apply the investigated methods to the numerical solution of the radial Schrodinger equation. 

T. E. Simos and P. Stability, Trigonometric-Fitting and the Numerical Solution of the Radial Schrodinger Equation, Computer Physics Communications, 2009, 180(7), 1072-1085. 

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