Coulomb Potential

The hydrogen atom therefore leads to a special case $Z=1$ of the solution of a stationary Schrödinger equation in the central potential

$${\color{col1}V(r) = - \frac{Ze^2}{4\pi \varepsilon_0 r}.}$$ (1.6)

Here, $Ze$ is introduced in order to be able to later generalise from proton charge $+e$ to arbitrary charge $Ze$. We use Dirac kets and write the stationary Schrödinger equation for $\hat{H}$

$$\hat{H} \vert\Psi\rangle = E \vert\Psi\rangle \leftrightarrow \left[- \frac{\hbar^2}{2m}\Delta + V(r)\right] \Psi({\bf r}) = E \Psi({\bf r})$$ (1.7)

with the Hamiltonian

$$\hat{H}=- \frac{\hbar^2}{2m}\Delta - \frac{Ze^2}{4\pi \varepsilon_0 r}.$$ (1.8)