 First excited state

A first excited state has the lowest allowed energy eigenvalue strictly above the ground-state energy, when such a next eigenvalue exists. The level can be degenerate, in which case several independent first excited states share that energy.

For a quantum harmonic oscillator, the ground state has occupation $n=0$ and the first excited state $n=1$, separated by $\hslash \omega$. In an infinite square well labeled by $n=1,2,\dots$, the first excited state instead has label $n=2$. “First” orders the energies rather than imposing a universal index convention. A system with a continuum starting at its lowest energy need not possess a normalizable first excited state at all.