Groundstate

In quantum mechanics, the ground state of a system is its lowest-energy state. It is the eigenstate of the system’s Hamiltonian H with the smallest eigenvalue E0, satisfying H|ψ0> = E0|ψ0>. If the Hamiltonian is time-independent, the ground-state wavefunction evolves only by a global phase e^{-iE0 t/ħ}, so observable properties remain constant in time. The ground-state energy E0 is bounded below, ensuring the system’s stability.

In many-body physics and quantum field theory, the term ground state often refers to the overall vacuum state—the lowest-energy configuration of all degrees of freedom. In condensed matter, the ground state encodes zero-temperature properties and can host quantum phases or order. Zero-point energy reflects that E0 may be nonzero even when all particles occupy their lowest motional states.

Common examples include the harmonic oscillator, whose ground-state energy is E0 = ħω/2 with a Gaussian wavefunction, and the hydrogen atom, whose ground state is the 1s orbital with energy −13.6 eV for the electron. Ground states may be unique or degenerate, depending on symmetries and interactions. Degeneracy can lead to phenomena such as spontaneous symmetry breaking when a system selects a particular ground state.

Finding the ground state typically involves solving the eigenproblem H|ψ> = E|ψ> for the lowest eigenvalue, guided by the variational principle. Numerical methods include exact diagonalization, density functional theory, density matrix renormalization group, and quantum Monte Carlo. The ground state determines zero-temperature observables and often underpins the phase structure and low-energy excitations of a system.