For a free particle, the Hamiltonian H = − − ℏ ℏ 2 2 m ∇ ∇ 2 is invariant under translations. Translation commutes with the Hamiltonian: [ H , T ^ ^ ] = 0 . However, if we express the Hamiltonian in the basis of the translation operator, we will find that H has doubly degenerate eigenvalues. It can be shown that to make the CSCO in this case, we need another operator called the parity operator Π Π , such that [ H , Π Π ] = 0 . { H , Π Π } forms a CSCO.