= Excited state
{wiki}
In the stationary-state terminology used here, an excited state is a normalized <energy eigenstate> with <energy eigenvalue> $E>E_0$, where $E_0$ is an attained <ground-state energy>. Its excitation cost above the <ground state> is $\Delta E=E-E_0>0$. Several <energy eigenstates> can share one excited <energy eigenvalue> through <degenerate energy levels>. An arbitrary <quantum superposition> with <energy> <expectation value> above $E_0$ need not be a single stationary <excited state>; its energy content is described by its <Born rule> probabilities or, for a continuum, its <scalar spectral measure>.
An <excited state> can be a <stationary state>: the second <infinite square well> mode is an immediate example. Spontaneous <photon emission> is a transition caused by coupling matter to the <electromagnetic field>; it is absent from an isolated matter-only <Hamiltonian operator>. The relevant <energy eigenstate> of that isolated model is therefore not automatically an <energy eigenstate> of the full interacting system. This distinction explains how an ideal stationary <excited state> can approximate a state with a finite experimental lifetime. See <electric-dipole interaction> for the coupling used in the simplest optical model.
For a one-photon transition between stationary internal states, neglecting recoil, <conservation of energy> requires
$$
h\nu=E_f-E_i\quad\text{for absorption},\qquad h\nu=E_i-E_f\quad\text{for emission}.
$$
Here $h$ is the <Planck constant> and $\nu$ is the <photon> <frequency>. Energy matching alone does not guarantee a transition: in the <electric-dipole interaction> approximation its strength also involves the <matrix element> $\langle f|\mathbf d\cdot\boldsymbol\epsilon|i\rangle$, where $\mathbf d$ is the <electric dipole moment> operator and $\boldsymbol\epsilon$ specifies <photon polarization>. A zero <matrix element> gives an <electric-dipole selection rule>. Thus an absent optical line does not prove the corresponding <energy eigenvalue> is absent; a different interaction can couple different states. https://farside.ph.utexas.edu/teaching/qmech/lectures/node121.html[Richard Fitzpatrick, Quantum Mechanics, Selection Rules]{ref}. The <Franck-Hertz experiment> gives a particularly instructive comparison between collision excitation and <photon emission>.
The <first excited state> terminology assumes a least <energy eigenvalue> above $E_0$. Such a next level is not guaranteed by <quantum mechanics>: a <continuous spectrum> can start immediately above the spectral bottom, or <energy eigenvalues> can accumulate there. For example, a <Hamiltonian operator> with <energy eigenvalues> $0,\varepsilon,\varepsilon/2,\varepsilon/3,\ldots$, where $\varepsilon>0$, has a <ground-state energy> zero but no least positive <energy eigenvalue>: every candidate $\varepsilon/n$ is exceeded downward by $\varepsilon/(n+1)$. A <free particle> on the whole line has neither a <ground state> nor a first stationary <excited state>. Each use of a discrete excitation ladder therefore needs a specified <Hamiltonian operator> and its <boundary conditions>.
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