The Radial Part

The radial part of the Schrödinger equation is obtained from (4.62) with $c=-l(l+1)$,

$$\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial... ...)}{\partial r}\right)+\frac{2m}{\hbar^2}[E-V(r)]R(r) -\frac{l(l+1)}{r^2}R(r)=0.$$ (290)

For the hydrogen atom, the attractive Coulomb potential generated by the proton (charge $+e>0$, $Z=1$) is

$$V(r)= -\frac{e^2}{4\pi\varepsilon_0 r}.$$ (291)

Strictly speaking, we are now dealing with a problem where many particles are involved: the electron and the proton which itself is composed of smaller elementary particles, the quarks. In such cases our single particle Schrödinger equation is no longer strictly valid. We neglect the inner structure of the proton and also use the fact that it is much heavier than the electron. As in the case of other two-body problems one can introduce center-of-mass and relative coordinates and reduce the problem to a one-particle problem. The mass $m$ is a reduced mass but it is very close to the electron mass.

Again, we do not explicitely solve for the possible energy eigenvalues $E$ and the radial eigenfunctions here but present the result: For bound states where the electron is bound to the attractive potential, the possible eigenvalues $E=E_n$ are labeled by a quantum number $n$,

$$E_n = -\frac{1}{2}\frac{e^2}{4\pi\varepsilon_0 a_0}\frac{1}{n^2},\quad n=1,2,3,...$$

$$a_0 := \frac{4\pi\varepsilon_0 \hbar^2}{me^2} .$$ (292)

The radial eigenfunctions for the bound states are

$$R_{nl}(r) = -\frac{2}{n^2}\sqrt{\frac{(n-l-1)!}{[(n+l)!]^3}} e^{-r/na_0}\left... ...r}{na_0}\right)^lL^{2l+1}_{n+l}\left(\frac{2r}{na_0}\right),\quad l=0,1,...,n-1$$ (293)

$$L^m_n(x) = (-1)^m\frac{n!}{(n-m)!}e^x x^{-m}\frac{d^{n-m}}{dx^{n-m}}e^{-x} x^n .$$

The wave functions of the bound states of the hydrogen atom (i.e. the attractive Coulomb potential (4.68)) are therefore given by the product of radial and angular part according to our separation ansatz (4.61),

$$\Psi_{nlm}(r,\theta,\varphi) = R_{nl}(r)Y_{lm}(\theta,\varphi).$$ (294)