Numerical Solution of the One-Dimensional Time-Independent Schrodinger Equation

4 NUMERICAL SOLUTION OF THE ONE-DIMENSIONAL TIME-INDEPENDENT SCHRODINGER EQUATION 

Don t be confused by the notation. The subscripts n — l,n, and n + 1 do not label different states but rather indicate values of one particular wave function ij/ and its derivatives at points on the x axis separated by the interval s. The n subscript means the functions are evaluated at the point x [Eq. (4.67)]. 

To get which occurs in (4.69), we replace/in (4.68) by ift and multiply the resulting equation by 5 this gives 

We shall neglect the s term in (4.72), since the s terms were neglected in earlier equations. Solving (4.72) for get 

FIGURE 4.7 The number of nodes in a Numerov-method solution as a function of the energy ., 

4 Numerical Solution of the One-Dimensional Time-Independent Schrodinger Equation 

We solved the Schrodinger equation exactly for the particle in a box and the harmonic oscillator. For many potential-energy functions y(x), the one-particle, one-dimensional Schrodinger equation cannot be solved exactly. This section presents a numerical method (the Numerov method) for computer solution of the one-particle, one-dimensional Schrodinger equation that allows one to get accurate bound-state eigenvalues and eigenfunctions for an arbitrary F(x). 

To solve the Schrodinger equation numerically, we deal with a portion of the x axis that includes the classically allowed region and that extends somewhat into the classically forbidden region at each end of the classically allowed region. We divide this portion of the X axis into small intervals, each of length s (F. 4.7). The points Xq and x x the endpoints of this portion, and x is the endpoint of the nth interval. Let Pn-u / n+1 denote the values of ip at the points x — s, x , and x -E s, respectively (these are the endpoints of adjacent intervals) 

FIGURE4.7 Kversusx fora one-particle, onedimensional system. 

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