The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. In quantum field theory, the ground state is usually called the vacuum. If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator that acts non-trivially on a ground state and commutes with the Hamiltonian of the system. According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature.
Absence of nodes in one dimension In one dimension, the ground state of the Schrödinger equation can be proven to have no nodes.
Derivation Consider the average energy of a state with a node at x = 0; i.e., ψ(0) = 0. The average energy in this state would be
⟨ ψ | H | ψ ⟩ = ∫ d x ( − ℏ 2 2 m ψ ∗ d 2 ψ d x 2 + V ( x ) | ψ ( x ) | 2 ) , {\displaystyle \langle \psi |H|\psi \rangle =\int dx\,\left(-{\frac {\hbar ^{2}}{2m}}\psi ^{*}{\frac {d^{2}\psi }{dx^{2}}}+V(x)|\psi (x)|^{2}\right),}
where V(x) is the potential. With integration by parts:
∫ a b ψ ∗ d 2 ψ d x 2 d x = [ ψ ∗ d ψ d x ] a b − ∫ a b d ψ ∗ d x d ψ d x d x = [ ψ ∗ d ψ d x ] a b − ∫ a b | d ψ d x | 2 d x {\displaystyle \int _{a}^{b}\psi ^{*}{\frac {d^{2}\psi }{dx^{2}}}dx=\left[\psi ^{*}{\frac {d\psi }{dx}}\right]_{a}^{b}-\int _{a}^{b}{\frac {d\psi ^{*}}{dx}}{\frac {d\psi }{dx}}dx=\left[\psi ^{*}{\frac {d\psi }{dx}}\right]_{a}^{b}-\int _{a}^{b}\left|{\frac {d\psi }{dx}}\right|^{2}dx}
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