= Ground state
{wiki}
A ground state is a normalized <energy eigenstate> that attains the lowest possible <energy> of a <quantum system>. Precisely, let the <Hamiltonian operator> $H$ be a <self-adjoint operator> bounded below on a <Hilbert space>, and write $E_*=\inf\sigma(H)$ for the <infimum> of its <spectrum>. A <ground state> is a <unit vector> $\psi_0$ in the <operator domain> $D(H)$ satisfying
$$
H\psi_0=E_*\psi_0,\qquad \|\psi_0\|=1.
$$
Thus a <ground state> exists exactly when the spectral bottom $E_*$ is an <energy eigenvalue>. The lowest <energy eigenvalue> among some listed <bound states> is insufficient if other parts of the <spectrum> lie below it. The <ground-state subspace> collects all minimizing <energy eigenstates>; it can have <ground-state degeneracy>. The <quantum variational principle> characterizes $E_*$ as the <infimum> of normalized <energy> <expectation values>. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf[Gerald Teschl, Mathematical Methods in Quantum Mechanics, Theorem 2.19]{ref}.
For the one-dimensional <quantum harmonic oscillator>, the <ground state> has <energy> $\hbar\omega/2$ above the minimum of the <potential energy>. This <zero-point energy> is compatible with being the lowest allowed <energy>: a localized <wavefunction> cannot simultaneously have zero <kinetic energy>. For a <free particle> on the whole line, the spectral bottom is zero but is unattained by any <normalizable wavefunction>; see <ground-state energy>. A lower bound therefore need not describe a physically preparable <energy eigenstate>.
The <ground state> supplies a reference for excitation costs and for the <quantum variational principle>. A <ground state> is also a <stationary state> under a time-independent <Hamiltonian operator>: its <quantum phase> changes, while its position <probability density> stays fixed. A nonzero <ground-state energy> does not imply motion along a classical trajectory, and <ground-state degeneracy> means there need not be a unique lowest <quantum state>.
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