04 - Energies and Wave Functions of Atom with Single Electron

Energies and Wave Functions of Atom with Single Electron

Consider atom of single electron orbiting nucleus of atomic number $Z$ then

$$E=E_n=-\frac{{\hslash }^2{Z}^2}{2M{a}^2}\cdot \frac{1}{N^2}$$

where $N\in \mathbb{N}$ is the principal quantum number

with corresponding wave functions

$$\psi ={\psi }_{N,l,m}(r,\theta ,\varphi )=f_{N,l}(r)e^{-Zr/aN}{Y}_{l,m}(\theta ,\varphi )$$

where $f_{N,l}(r)$ is a polynomial of degree $N-1$ (generalised Laguerre polynomials)

Wave functions have $L^2$-eigenvalue $l(l+1){\hslash }^2$ and $L_3$-eigenvalue $m\hslash$
where $l$ is the azimuthal quantum number and $m$ is magnetic quantum number

Degeneracy of $E_N$

$$\text{degeneracy of }{E}_N=\sum _{l=0}^{N-1}(2l+1)=N^2$$

where ground state $N=1$ is non-degenerate so a stable atom