Just like the matrix elements of X + iP and X − iP for the harmonic oscillator Hamiltonian, this commutation law implies that these operators only have certain off diagonal matrix elements in states of definite m, L z ( ( X + i Y ) | m ⟩ ⟩ ) = ( X + i Y ) L z | m ⟩ ⟩ + ( X + i Y ) | m ⟩ ⟩ = ( m + 1 ) ( X + i Y ) | m ⟩ ⟩ meaning that the matrix (X + iY) takes an eigenvector of L z with eigenvalue m to an eigenvector with eigenvalue m + 1. Similarly, (X − iY) decrease m by one unit, while Z does not change the value of m.