Since U is nontrivial, at least one pair of | a ⟩ ⟩ and U | a ⟩ ⟩ must represent distinct states. Therefore, H has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.