The most important categorization of the solution to the hydrogen atom is the principle quantum number, n. We regard this as the most important because this number is related to the energy associated with that particular wavefunction. For hydrogen-like atoms (one electron and a nucleus) all solutions with the same n value are degenerate (they have the same energy). The energy of the level in joules is given by
\[E_n = -2.18\times 10^{-18} \left({Z^2 \over n^2}\right)\]
where Z is the nuclear charge. For hydrogen (Z=1), the energy levels start at n=1 with a value of -2.18 x 10-18J. Then energies are negative as we have defined E=0 as the energy of the electron and the nucleus at a distance of infinity. The electron is more stable (lower in energy) when it is closer to the nucleus. Since we have defined the separated nucleus and electron as zero, the more stable energies must all be negative. The more negative, the more stable it is. Then they move up in energy (closer to zero) with a spacing that decreases with each level until they all approach an energy of zero (n=infinity). The energy also depends on the square of the nuclear charge, Z. The ground state energy (n=1) for helium (1s2, Z=2) is 4 times lower than the ground state energy (n=1) for hydrogen (1s1, Z=1).
We can now use this formula to find the energy difference between any two states in the hydrogen atom. This will come out to be identical to the trends that were found for the Rhydberg equation.
One interesting application of this formula is that we can use it to find the energy required to remove the electron from a hydrogen-like atom. This is the energy difference between the ground state (lowest energy) \( n = 1\) and the final state in which the electron is separated from the nucleus. The electron separated from the nuclear corresponds to \( n = \infty \) . The energy for \( n = \infty \) is zero.