Ground state

A ground state is an energy eigenstate with the smallest possible energy. For noninteracting particles, the many-body ground state is obtained by filling one-particle levels subject to the particles' exchange statistics.

The ground-state subspace is the eigenspace of a Hamiltonian operator at its lowest eigenvalue $E_0$. A computational history state construction can have many valid quantum witnesses, hence a degenerate ground space. Perturbation bounds must optimize on this entire subspace, rather than on a chosen basis of ground states.

The ground-state degeneracy counts independent states minimizing the energy. For a classical continuous-symmetry system it can instead be described by a manifold of minimizing configurations, such as the sphere of orientations of a Néel state in an isotropic Heisenberg antiferromagnet.

The ground-state energy is the smallest energy eigenvalue of a Hamiltonian operator.

For every normalized trial state $\psi$ in the domain of a Hamiltonian operator $H$ that is bounded below,where $E_0$ is the ground-state energy. Equality holds precisely when the trial state lies in the ground-state eigenspace.

For $V=-A{e}^{-\mu r}/r$, the normalized exponential with scale $\alpha$ gives $E(\alpha )={\hslash }^2{\alpha }^2/(2m)-4A{\alpha }^3/(\mu +2\alpha {)}^2$. Choosing $\alpha =\mu /2$ proves binding if $\mu <Am/{\hslash }^2$. Failure to obtain a negative trial expectation outside this range does not prove absence of a bound state. At equality a small positive admixture of the scale-$\mu$ exponential has a negative cross matrix element and proves binding too.

For a bound stationary state in a homogeneous potential $V(\lambda x)={\lambda }^{n}V(x)$, scaling a normalized wavefunction gives stationarity of ${\lambda }^2⟨T⟩+{\lambda }^{-n}⟨V⟩$ at $\lambda =1$. Hence $2⟨T⟩=n⟨V⟩$.