In the correspondence limit, when indices m, n are large and nearby, while k, r are small, the rate of change of the matrix elements in the diagonal direction is the matrix element of the J derivative of the corresponding classical quantity. So its possible to shift any matrix element diagonally through the correspondence, A ( m + r ) ( n + r ) − − A m n ≈ ≈ r ( d A d J ) m n where the right hand side is really only the (m − n)'th Fourier component of dA / dJ at the orbit near m to this semiclassical order, not a full well-defined matrix.