The energy of the hydrogen atom is quantized. Only those energies given by Equation 41.2 allow the wave function to satisfy the boundary conditions. The allowed energies of hydrogen depend only on the principal quantum number $n$, but for other atoms the energies will depend on both $n$ and $l$. In anticipation of using both quantum numbers, FIGURE 41.3 is an energy-level diagram for the hydrogen atom in which the rows are labeled by $n$ and the columns by $l$. The left column contains all of the $l=0s$ states, the next column is the $l=1p$ states, and so on.

FIGURE 41.3 Energy-level diagram for the hydrogen atom.

Because the quantum condition of Equation 41.3 requires $n>l$, the $s$ states begin with $n=1$, the $p$ states begin with $n=2$, and the $d$ states with $n=3$. That is, the lowest-energy $d$ state is $3d$ because states with $n=1$ or $n=2$ cannot have $l=2$. For hydrogen, where the energy levels do not depend on $l$, the energy-level diagram shows that the $3s$, $3p$, and $3d$ states have equal energy. Figure 41.3 shows only the first few energy levels for each value of $l$, but there really are an infinite number of levels, as $n\to \infty$, crowding together beneath $E=0$. The dashed line at $E=0$ is the atom’s ionization limit, the energy of a hydrogen atom in which the electron has been moved infinitely far away to form an $\text{H}{}^{+}$ ion.

The lowest energy state, the $1s$ state with $E_1=-13.60\text{eV,}$ is the ground state of hydrogen. The value $\left|E_1\right|=13.60\text{eV}$ is the ionization energy, the minimum energy that would be needed to form a hydrogen ion by removing the electron from the ground state. All of the states with $n>1$ (i.e., the states with energy higher than the ground state) are excited states.