Hydrogen-like atom

Hydrogen-like atom has nucleus and one electron. C⁵⁺ ion, for example, has one electron and nucleus with charge Z = +6.

Central potential of nucleus

1. $${V^{NUC}}\left( r \right) =  - \frac{Z}{r}$$

affects on electron states, where r is the distance from the nucleus.

Please note, that all formulas we are writing in atomic units (a.u.) where

2. $$\begin{array}{l}
   e = 1\\
   \hbar  = 1\\
   {m_e} = 1\\
   4\pi {\varepsilon _0} = 1
   \end{array}$$

For spherical symmetry we can look for the solution of the Schrödinger equation in the form

3. $$\Psi \left( {r,\theta ,\varphi } \right) = \Psi \left( r \right)Y\left( {\theta ,\varphi } \right)$$

where $Y\left( {\theta ,\varphi } \right)$ is spherical harmonics. Moreover, the Schrödinger equation for spherical symmetry can be separated on two independent equations for $r$ and ${\theta ,\varphi }$. 

The Schrodinger equation in spherical coordinates for part depending on $r$ only, the radial Schrodinger equation is

4. $$\left( { - \frac{1}{{2r}}\frac{{{d^2}}}{{d{r^2}}}r + \frac{{l\left( {l + 1} \right)}}{{2{r^2}}} - \frac{Z}{r}} \right){\Psi _{nl}}\left( r \right) = {E_{nl}}{\Psi _{nl}}\left( r \right)$$

The lowest energy state of H-like atom is $1s$ state with $n=1$, $l=0$, where $n$ is the principal quantum number and $l$ is the angular quantum number. This $1s$ state has spherical symmetry and wave function of this state does not depend on spherical angles

5. $${\Psi _{1s}}\left( {r,\theta ,\varphi } \right) = {\Psi _{1s}}\left( r \right)$$

For $1s$ state the exact analytical solution [1] of Eq. (4)

6. $$\begin{array}{l}
   {\Psi _{10}}\left( r \right) = \frac{1}{{\sqrt \pi }}{Z^{\frac{3}{2}}}{e^{ - Zr}}\\
   {E _{10}} =  - \frac{{{Z^2}}}{2}
   \end{array}$$

References:

[1] Pauling, L.; Wilson, E.B. Introduction to Quantum Mechanics, McGraw-Hill, New York, 1935.

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