The operator "X" is a "raising operator" for "N" if "c" is real and positive, and a "lowering operator" for "N" if "c" is real and negative. If "N" is a Hermitian operator then "c" must be real and the Hermitian adjoint of "X" obeys the commutation relation:  In particular, if "X" is a lowering operator for "N" then "X"† is a raising operator for "N" and vice versa. A particular application of the ladder operator concept is found in the quantum mechanical treatment of angular momentum. For a general angular momentum vector, J, with components, "Jx", "Jy" and "Jz" we define the two ladder operators, "J+" and "J–":  where "i" is the imaginary unit. The commutation relation between the cartesian components of "any" angular momentum operator is given by  where "εijk" is the Levi-Civita symbol and each of "i", "j" and "k" can take any of the values "x", "y" and "z". From this the commutation relations between the ladder operators and "Jz" can easily be obtained:  The properties of the ladder operators can be determined by observing how they modify the action of the "Jz" operator on a given state:  Compare this result with:  Thus we conclude that formula_16 is some scalar multiplied by formula_17,  This illustrates the defining feature of ladder operators in quantum mechanics: the incrementing (or decrementing) of a quantum number, thus mapping one quantum state onto another. This is the reason that they are often known as raising and lowering operators. To obtain the values of "α" and "β" we first take the norm of each operator, recognizing that "J"+ and "J"− are a Hermitian conjugate pair (formula_20),  The product of the ladder operators can be expressed in terms of the commuting pair "J"2 and "Jz",  Thus we can express the values of |"α"|2 and |"β"|2 in terms of the eigenvalues of "J"2 and "Jz",  The phases of "α" and "β" are not physically significant, thus they can be chosen to be positive and real (Condon-Shortley phase convention).