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Year 2 of the mathematics course of the University of Oxford

Quantum theory course of the University of Oxford

Past exam of the quantum theory course of the University of Oxford

2014

Substitute the <power series> and collect each power of $\xi$:
$$
\sum_{k\geq0}\left[(k+2)(k+1)a_{k+2}+(\epsilon-1-2k)a_k\right]\xi^k=0.
$$
Matching the <power series> coefficients gives the <recurrence relation>
$$
\boxed{a_{k+2}=\frac{2k+1-\epsilon}{(k+2)(k+1)}a_k,\qquad k=0,1,2,\ldots.}
$$
The <even> and <odd> coefficients form separate sequences, determined by $a_0$ and $a_1$, respectively. For example, $a_2=(1-\epsilon)a_0/2$ and $a_3=(3-\epsilon)a_1/6$. These are the two independent <power series solutions of the Hermite differential equation>.

Solution

Past exam of the quantum theory course of the University of Oxford / 2014 / 2 / b / Solution

Past exam of the quantum theory course of the University of Oxford / 2014 / 2 / b / Solution

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