Energy Levels of the Hydrogen Atom

Total mechanical energy in an orbit. An electron in the $n$th Bohr orbit of a hydrogen-like atom has both kinetic energy (from its orbital motion) and electrostatic potential energy (from its attraction to the nucleus); its total energy $E_n$ is the sum of the two.

Kinetic energy. From the force-balance condition of Section 1.4 ($\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r^2}=\frac{mv^2}{r}$), rearranged as $mv^2 = \frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r}$, the kinetic energy is

Potential energy. Taking electrostatic potential energy to be zero when the electron is infinitely far from the nucleus (the standard convention), the potential energy at separation $r$ from an attractive Coulomb force between charges $+Ze$ and $-e$ is

negative, since the two charges attract and work must be done to pull them apart.

Total energy. Adding the two:

Notice the total energy is exactly the NEGATIVE of the kinetic energy alone, and exactly half the magnitude of the potential energy -- a general feature of any inverse-square-force circular orbit (the same relation holds for a planet in a circular orbit under gravity). The overall negative sign shows the electron is BOUND to the nucleus: energy would have to be SUPPLIED to remove it to $r=\infty$ (Numerical 9 uses exactly this idea to find an ionisation energy).

Substituting the orbit radius. Using $r_n = \dfrac{\epsilon_0n^2h^2}{\pi mZe^2}$ from Section 1.4.1:

Carrying out this substitution numerically for $Z=1$ (hydrogen) gives the standard, compactly written result

$\boxed{E_n = -\frac{13.6\,Z^2}{n^2}\ \text{eV}}$ …