Power-Series Solution of Differential Equations

So far we have considered only cases where the potential energy y(jc) is a constant. This makes the Schrodinger equation a second-order linear homogeneous differential equation with constant coefficients, which we know how to solve. For cases in which V varies with X, a useful approach is to try a power-series solution of the Schrodinger equation. 

Section 2.2 [Eqs. (2.10) and (4.1) are the same], we get trigonometric solutions when the roots of the auxiliary equation are pure imaginary  

Now let us solve (4.1) using the power-series method. We start by assuming that the solution can be expanded in a Taylor series (see Prob. 4.1) about x = 0 that is, we assume that 

We want to combine the two sums in (4.7). Provided certain conditions are met, we can add two infinite series term by term to get their sum  

The last equality in (4.9) is valid because the summation index is a dummy variable it makes no difference what letter we use to denote this variable. For example, the sums 2 =i and Sm=i are equal because only the dummy variables in the two sums differ. This equality is easy to see if we write out the sums  

Because only the dummy variables in the two sums differ, the sums are equal. This is easy to see if we write them out  

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Section 4.1 Power-Series Solution of Differential Equations 63... 

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