Radial Solutions

The solutions of Eq. (II.1.7) are now seperated into radial part $R_{nl}(r)$ and spherical part $Y_{lm}(\theta,\varphi)$,

$$\Psi_{nlm}(r,\theta,\varphi) = R_{nl}(r)Y_{lm}(\theta,\varphi),$$ (1.17)

where radial eigenfunctions for the bound states are characterised by the two integer quantum numbers $n\ge l+1$ and $l$,

$$R_{nl}(r) = -\frac{2}{n^2}\sqrt{\frac{(n-l-1)!}{[(n+l)!]^3}} e^{-Zr/na_0}\lef... ...}{na_0}\right)^lL^{2l+1}_{n+l}\left(\frac{2Zr}{na_0}\right),\quad l=0,1,...,n-1$$ (1.18)

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

The radial wave functions $R_{nl}(r)$ have $n-l$ nodes. For these states, the possible eigenvalues only depend on $n$, $E=E_n$ with

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

$$a_0 \equiv \frac{4\pi\varepsilon_0 \hbar^2}{me^2} .$$ (1.19)

In Dirac notation, we write the stationary states as $\vert nlm\rangle$ with the correspondence

$$\vert nlm\rangle \leftrightarrow \langle {\bf r}\vert nlm\rangle \equiv \Psi_{nlm}({\bf r}).$$ (1.20)

The ground state is $\vert GS\rangle = \vert 100\rangle$ with energy $E_0=-13.6$ eV. The degree of degeneracy of the energy level $E_n$, i.e. the number of linearly independent stationary states with quantum number $n$, is

$$\sum_{l=0}^{n-1}(2l+1) = n^2$$ (1.21)

Backup literature: lecture notes QM 1

http://brandes.phy.umist.ac.uk/QM/http://brandes.phy.umist.ac.uk/QM/, textbooks Merzbacher [2], Landau-Lifshitz III [1], Gasiorowisz [3].