whole universe, is conserved. However, L and S are not generally conserved. For example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total J remaining constant. Commutation relations Commutation relations between components The orbital angular momentum operator is a vector operator, meaning it can be written in terms of its vector components . The components have the following commutation relations with each other: where denotes the commutator This can be written generally as where l, m, n are the component indices (1 for x, 2 for y, 3 for z), and denotes the Levi-Civita symbol. A compact expression as one vector equation is also possible: The commutation relations can be proved as a direct consequence of the canonical commutation relations , where is the Kronecker delta. There is an analogous relationship in classical physics: where Ln is a component of the classical angular momentum operator, and is the Poisson bracket. The same commutation relations apply for the other angular momentum operators (spin and total angular momentum): These can be assumed to hold in analogy with L. Alternatively, they can be derived as discussed below. These commutation relations mean that L has the mathematical structure of a Lie algebra, and the are its structure constants. In this case, the Lie algebra is SU(2) or SO(3) in physics notation ( or respectively in mathematics notation), i.e. Lie algebra associated with rotations in three dimensions. The same is true of J and S. The reason is discussed below. These commutation relations are relevant for measurement and uncertainty, as discussed further below. In molecules the total angular momentum F is the sum of the rovibronic (orbital) angular momentum N, the electron spin angular momentum S, and the nuclear spin angular momentum I. For electronic singlet states the rovibronic angular momentum is denoted J rather than N. As explained by Van Vleck, the components of the molecular rovibronic angular momentum referred to molecule-fixed axes have different commutation relations from those given above which are for the components about space-fixed axes. Commutation relations involving vector magnitude Like any vector, the square of a magnitude can be defined for the orbital angular momentum operator, is another quantum operator. It commutes with the components of , One way to prove that these operators commute is to start from the [Lℓ, Lm] commutation relations in the previous section: Mathematically, is