Principle quantum number

The principal quantum number, symbolised at \[n\] is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Electrons with a higher \[n\] value will have a higher energy as evidenced below:
\[E_{n}=\frac{E_{1}}{n^{2}},\,n=1,2,3\dotso\]
For example, the bound state energies of the electron in a hydrogen atom is:
\[\frac{-13.6eV}{n^{2}},\,n=1,2,3\dotso\]
This formula is ideal for single-electron systems as the only significant interactions are between the single electron and the nucleus, whereas for atoms like Xe, it involves much more complex interactions such as electron shielding and interelectron repulsion.

Since Bohr's model involved only a single electron, it could also be applied to the single electron ions \[\ce{H^{+}}\], \[\ce{He^{2+}}\], \[\ce{Li^{3+}}\], and so forth, which differ from hydrogen only in their nuclear charges, and so one-electron atoms and ions are collectively referred to as hydrogen-like atoms. The energy expression for hydrogen-like atoms is a generalization of the hydrogen atom energy, in which \[Z\] is the nuclear charge (+1 for hydrogen, +2 for He, +3 for Li, and so on) and \[k\] is the Rydberg unit of energy in joules.
\[E_{n}=-\frac{kZ^{2}}{n^{2}},\,n=1,2,3\dotso\]

The relation between the Azimuthal quantum number, \[\ell\] and principle quantum number is that the possible values of \[\ell\] is \[\ell=0\dotso n-1\]. For example, for \[n=3\] the possible values for \[\ell\] is 0, 1 and 2 which correspond to the 3s, 3p and 3d shells.