Frobenius method

In Chapters 4, 5, and 6 the Schrodinger equation is applied to three systems the harmonie oseillator, the orbital angular momentum, and the hydrogen atom, respectively. The ladder operator technique is used in each case to solve the resulting differential equation. We present here the solutions of these differential equations using the Frobenius method. 

This equation can be solved by the Frobenius method, assuming a series solution... 

Determine the roots of the indidal relationship for the Frobenius method applied to the following equation ... 

There are two procedures available for solving this differential equation. The older procedure is the Frobenius or series solution method. The solution of equation (4.17) by this method is presented in Appendix G. In this chapter we use the more modem ladder operator procedure. Both methods give exactly the same results. 

The eigenvalues and eigenfimctions of the orbital angular momentum operator may also be obtained by solving the differential equation I ip = Xh ip using the Frobenius or series solution method. The application of this method is presented in Appendix G and, of course, gives the same results... 

Equation (6.24) may be solved by the Frobenius or series solution method as presented in Appendix G. However, in this chapter we employ the newer procedure using ladder operators. 

The Frobenius or series solution method for solving equation (G.l) assumes that the solution may be expressed as a power series in x... 

The MNM method further modifies the Hessian matrix based on the PD scheme, and it guarantees the matrix to be positive semidefinite. This process is fundamentally equivalent to modifying the molecular interactions defined in the original force field. In MNM, all PD s are changed to their nearest (in terms of the Frobenius norm) SPSD matrices ffi. 

For regular singular points, a series solution of the differential equation can be obtained by the method of Frobenius. This is based on the following generalization of the power series expansion ... 

Linear homogeneous equations containing nonconstant coefficients were not treated, except for the elementary Euler-Equidimensional equation, which was reduced to a constant coefficient situation by letting x = exp(r). In the next chapter, we deal extensively with the nonconstant coefficient case, starting with the premise that all continuous solutions are in fact representable by an infinite series of terms, for example ejqj (jc) = 1 + x + x /2 + x /3 + . This leads to a formalized procedure, called the Method of Frobenius, to find all the linearly independent solutions of homogeneous equations, even if coefficients are nonconstant. 

---