 We can define the transformation of the state from the state at time t to the state at time formula_22 as

To conserve energy in the wave we require

where formula_25 is the adjoint of U, the complex conjugate transpose of the matrix.This implies that a transformation that conserves energy must obey

where I is the identity operator and U is called a unitary operator.The unitary property is necessary to ensure energy conservation in state transformations.