a Casimir invariant of the Lie algebra SO(3) spanned by . As above, there is an analogous relationship in classical physics: where is a component of the classical angular momentum operator, and is the Poisson bracket. Returning to the quantum case, the same commutation relations apply to the other angular momentum operators (spin and total angular momentum), as well, Uncertainty principle In general, in quantum mechanics, when two observable operators do not commute, they are called complementary observables. Two complementary observables cannot be measured simultaneously; instead they satisfy an uncertainty principle. The more accurately one observable is known, the less accurately the other one can be known. Just as there is an uncertainty principle relating position and momentum, there are uncertainty principles for angular momentum. The Robertson–Schrödinger relation gives the following uncertainty principle: where is the standard deviation in the measured values of X and denotes the expectation value of X. This inequality is also true if x, y, z are rearranged, or if L is replaced by J or S. Therefore, two orthogonal components of angular momentum (for example Lx and Ly) are complementary and cannot be simultaneously known or measured, except in special cases such as . It is, however, possible to simultaneously measure or specify L2 and any one component of L; for example, L2 and Lz. This is often useful, and the values are characterized by the azimuthal quantum number (l) and the magnetic quantum number (m). In this case the quantum state of the system is a simultaneous eigenstate of the operators L2 and Lz, but not of Lx or Ly. The eigenvalues are related to l and m, as shown in the table below. Quantization In quantum mechanics, angular momentum is quantized – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where is reduced Planck constant: Derivation using ladder operators A common way to derive the quantization rules above is the method of ladder operators. The ladder operators for the total angular momentum are defined as: Suppose is a simultaneous eigenstate of and (i.e., a state with a definite value for and a definite value for ). Then using the commutation relations for the components of , one can prove that each of the states and is either zero or a simultaneous eigenstate of and , with