Chapter 13.7, Problem 22P

Find the energy eigenvalues and eigen functions for the hydrogen atom. The potential energy is $V\left(r\right)=-e^2/r$ in Gaussian units, where $e$ is the charge of the electron and $r$ is in spherical coordinates. Since $V$ is a function of $r$ only, you know from Problem 18 that the eigen functions are $R(r)$ times the spherical harmonics $Y_l^m\left(\theta ,\varphi \right)$, so you only have to find $R(r)$. Substitute $V\left(r\right)$ into the $R$ equation in Problem 18 and make the following simplifications: Let $x=2r/\alpha$, $y=rR$; show that then

$r=\alpha x/2,R\left(r\right)=\frac{2}{\alpha x}y\left(x\right),\frac{d}{dr}=\frac{2}{\alpha }\frac{d}{dx},\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)=\frac{2}{\alpha }x{y}^{″}.$

Let ${\alpha }^2=-2ME/h^2$ (note that for a bound state, $E$ is negative, so ${\alpha }^2$ is positive) and $\lambda =M{e}^2\alpha /h^2$, to get the first equation in Problem 22.26 of Chapter 12. Do this problem to find $y\left(x\right)$, and the result that $\lambda$ is an integer, say $n$. (Caution: not the same $n$ as in equation (22.26)). Hence find the possible values of $\alpha$ (these are the radii of the Bohr orbits), and the energy eigenvalues. You should have found $\alpha$ proportional to $n$; let $\alpha =na$, where $a$ is the value of $\alpha$ when $n=1$, that is, the radius of the first Bohr orbit. Write the solutions $R\left(r\right)$ by substituting back $y=rR$, and $x=2r/\left(na\right)$, and find $E_n$ from $\alpha$.