Chapter 12.22, Problem 27P

The functions which are of interest in the theory of the hydrogen atom are $f_n(x)=x^{l+1}{e}^{-x/2n}{L}_{n-l-1}^{2l+1}\left(\frac{x}{n}\right)$ where $n$ and $l$ are integers with $0\le l\leqq n-1$.

(Note that here $k=2l+1$, and we have replaced $n$ by $n-l-1;$ in this problem $L_2^3,$ say, means $l=1,n=4.$ ) For $l=1,$ show that $f_2(x)=x^2{e}^{-x/4},f_3(x)=x^2{e}^{-x/6}\left(4-\frac{x}{3}\right),f_4(x)=x^2{e}^{-x/8}\left(10-\frac{5x}{4}+\frac{x^2}{32}\right)$

Hint: Find the polynomials $L_0^3,L_1^3,L_2^3$ as in Problem 20 (with $k=3$ ) and then replace $x$ by $x/n.$ The functions $f_n(x)$ are very different from those in (22.29) since $x/n$ changes from one function to the next. However, it can be shown (Prob-lem 23.25) that for one fixed $l$, the set of functions $f_n(x),n\ge l+1,$ is an orthogonal set on $(0,\infty ).$ Verify this for these three functions. Hint: The integrals are $\Gamma$ functions see Chapter 11, Section 3.