 Solution

ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2014/2/b/solution

Substitute the power series and collect each power of

ξ

\sum_{k \geq 0} [(k + 2) (k + 1) a_{k + 2} + (ϵ - 1 - 2 k) a_{k}] ξ^{k} = 0.

a_{k + 2} = \frac{2 k + 1 - ϵ}{( k + 2 ) ( k + 1 )} a_{k}, k = 0, 1, 2, \dots .

The even and odd coefficients form separate sequences, determined by

a_{0}

and

a_{1}

, respectively. For example,

a_{2} = (1 - ϵ) a_{0} /2

and

a_{3} = (3 - ϵ) a_{1} /6

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