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

where radial eigenfunctions for the bound states are characterised by the two integer quantum numbers $ n\ge l+1$ and $ l$,
$\displaystyle R_{nl}(r)$ $\displaystyle =$ $\displaystyle -\frac{2}{n^2}\sqrt{\frac{(n-l-1)!}{[(n+l)!]^3}}
e^{-Zr/na_0}\lef...
...}{na_0}\right)^lL^{2l+1}_{n+l}\left(\frac{2Zr}{na_0}\right),\quad
l=0,1,...,n-1$ (1.18)
$\displaystyle L^m_n(x)$ $\displaystyle =$ $\displaystyle (-1)^m\frac{n!}{(n-m)!}e^x x^{-m}\frac{d^{n-m}}{dx^{n-m}}e^{-x} x^n$   generalized Laguerre polynomials$\displaystyle .$  

The radial wave functions $ R_{nl}(r)$ have \bgroup\color{col1}$ n-l$\egroup nodes. For these states, the possible eigenvalues only depend on \bgroup\color{col1}$ n$\egroup, \bgroup\color{col1}$ E=E_n$\egroup with
$\displaystyle E_n$ $\displaystyle =$ $\displaystyle -\frac{1}{2}\frac{Z^2e^2}{4\pi\varepsilon_0 a_0}\frac{1}{n^2},\quad n=1,2,3,...$   Lyman Formula  
$\displaystyle a_0$ $\displaystyle \equiv$ $\displaystyle \frac{4\pi\varepsilon_0 \hbar^2}{me^2}$   Bohr Radius$\displaystyle .$ (1.19)

In Dirac notation, we write the stationary states as \bgroup\color{col1}$ \vert nlm\rangle$\egroup with the correspondence
$\displaystyle \vert nlm\rangle \leftrightarrow \langle {\bf r}\vert nlm\rangle \equiv \Psi_{nlm}({\bf r}).$     (1.20)

The ground state is \bgroup\color{col1}$ \vert GS\rangle = \vert 100\rangle$\egroup with energy \bgroup\color{col1}$ E_0=-13.6$\egroup eV. The degree of degeneracy of the energy level \bgroup\color{col1}$ E_n$\egroup, i.e. the number of linearly independent stationary states with quantum number \bgroup\color{col1}$ n$\egroup, is
$\displaystyle \sum_{l=0}^{n-1}(2l+1) = n^2$     (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].


next up previous contents index
Next: A `Mini-Molecule': Perturbation Theory Up: Hydrogen Atom (non-relativistic) Previous: Orbital Angular Momentum   Contents   Index
Tobias Brandes 2005-04-26