Hydrogen Wavefunctions

All of the hydrogen wavefunctions you could ever want - All in one place!

Complete Wavefunctions

These functions combine the radial, angular, and time dependencies into one equation.

To find the time-independent solutions, remove the $e^{-i\frac{E_n}{\hbar}t}$.

To use atomic units (in terms of the Bohr radius), let $\frac{r}{a_0} = r$.

Of course, hydrogen itself has a nuclear charge of $Z=1$.

The wavefunctions have three quantum numbers, $n$, $l$, and $m_l$. They are parameters to the wavefunction $\Psi_{n,l,m_l}$.

Complex-Valued Wavefunctions

\begin{equation*} \Psi_{1,0,0}(r,\theta,\phi,t) = \frac{1}{\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{\frac{3}{2}}e^{\frac{-Zr}{a_0}}e^{-i\frac{E_{1}}{\hbar}t} \end{equation*}\begin{equation*} \Psi_{2,0,0}(r,\theta,\phi,t) = \frac{1}{4\sqrt{2\pi}}\left(\frac{Z}{a_0}\right)^{\frac{3}{2}}\left(2-\frac{Zr}{a_0}\right)e^{\frac{-Zr}{2a_0}}e^{-i\frac{E_{2}}{\hbar}t} \end{equation*}\begin{equation*} \Psi_{2,1,0}(r,\theta,\phi,t) = \frac{1}{4\sqrt{2\pi}}\left(\frac{Z}{a_0}\right)^{\frac{5}{2}}re^{\frac{-Zr}{2a_0}}\cos\theta e^{-i\frac{E_{2}}{\hbar}t} \end{equation*}\begin{equation*} \Psi_{2,1,\pm1}(r,\theta,\phi,t) = \frac{1}{8\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{\frac{5}{2}}re^{\frac{-Zr}{2a_0}}\sin\theta e^{\pm i\phi}e^{-i\frac{E_{2}}{\hbar}t} \end{equation*}

Real-Valued Wavefunctions

All of the $m_l=0$ wavefunctions above are already real-valued, so here I only list the new wavefunctions which combine $\pm m_l\neq0$.

Radial Wavefunctions

Angular Wavefunctions