Hydrogen Atom

Topics

• Spectrum Energy

$V(r)=-\frac{e^2}{4\pi {\epsilon }_0}\frac{1}{r}$

• This is the potential for an electron surrounding a hydrogen atom

$E_n=-\left[\frac{m_e}{2{\hslash }^2}{\left(\frac{e^2}{4\pi {\epsilon }_0}\right)}^2\right]\frac{1}{n^2}=\frac{E_1}{n^2},n\in {\mathbb{Z}}^{+}$

• Allowed energy levels of a

$E_1=\left[\frac{m_e}{2{\hslash }^2}{\left(\frac{e^2}{4\pi {\epsilon }_0}\right)}^2\right]=-13.6\mathrm{eV}$

• Equivalent to binding energy of hydrogen atom to electron/ionization energy

${\psi }_{100}(r,\theta ,\varphi )=\frac{1}{\sqrt{\pi a^3}}{e}^{-r/a}$

• Ground state of hydrogen

${\psi }_{nlm}(r,\theta ,\varphi )=R_{nl}(r)Y_l^m(\theta ,\varphi )$

• Wave function of a hydrogen atom for quantum numbers $n$, $l$, and , $m$
• $R_{nl}$ is given in the equation below and is the radial wave function for the hydrogen atom
• $Y_l^m(\theta ,\varphi )$ is the Spherical Harmonics

$R_{nl}(r)=\sqrt{{\left(\frac{2}{n{a}_0}\right)}^3\frac{(n-l-1)!}{2n(n+l)!}}{e}^{-r/n{a}_0}{\left(\frac{2r}{n{a}_0}\right)}^l\left[L_{n-l-1}^{2l+1}\left(\frac{2r}{n{a}_0}\right)\right]$

• The first term under the square root is a normalization factor
• $a_0$ is the Bohr radius
• $L_{n-l-1}^{2l+1}$ is the associated Laguerre polynomial

$R_{nl}(r)$ Values