# Quantum Degeneracy

In quantum mechanics, an energy level is said to be degenerate if the energy corresponds to two or more states of a quantum system. Also, two or more states of a quantum system are said to be degenerate if they give the same energy upon measurement. The set of all degenerate states of a quantum system that correspond a particular energy $E$ forms a (Hilbert) subspace, called the eigenspace of $E$. To see this, let $H$ be a Hamiltonian and $|\psi_1\rangle$, $|\psi_2\rangle$ two linearly independent eigenstates corresponding to the same eigenvalue (energy) $E$. Then
\begin{align*} H|\psi_1\rangle&=E|\psi_1\rangle\\ H|\psi_2\rangle&=E|\psi_2\rangle \end{align*}
Let $|\psi\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle$, a linear combination (superposition) of $|\psi_1\rangle$ and $|\psi_2\rangle$. Then
\begin{align*} H|\psi\rangle&=H(c_1|\psi_1\rangle+c_2|\psi_2\rangle)\\ &=c_1H|\psi_1\rangle+c_2H|\psi_1\rangle\\&=c_1E|\psi_1\rangle+c_2E|\psi_2\rangle\\ &=E(c_1|\psi_1\rangle+c_2|\psi_2)\\ &=E|\psi\rangle \end{align*}
The dimension of the eigenspace corresponding to an eigenvalue (energy) $E$ is called the degree of degeneracy of $E$.