the commutation relations, the possible eigenvalues can be found. These eigenstates are denoted where is the angular momentum quantum number and is the angular momentum projection onto the z-axis. They comprise the spherical basis, are complete, and satisfy the following eigenvalue equations, The raising and lowering operators can be used to alter the value of , where the ladder coefficient is given by: In principle, one may also introduce a (possibly complex) phase factor in the definition of . The choice made in this article is in agreement with the Condon–Shortley phase convention. The angular momentum states are orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and are assumed to be normalized, Here the italicized and denote integer or half-integer angular momentum quantum numbers of a particle or of a system. On the other hand, the roman , , , , , and denote operators. The symbols are Kronecker deltas. Tensor product space We now consider systems with two physically different angular momenta and . Examples include the spin and the orbital angular momentum of a single electron, or the spins of two electrons, or the orbital angular momenta of two electrons. Mathematically, this means that the angular momentum operators act on a space of dimension and also on a space of dimension . We are then going to define a family of "total angular momentum" operators acting on the tensor product space , which has dimension . The action of the total angular momentum operator on this space constitutes a representation of the su(2) Lie algebra, but a reducible one. The reduction of this reducible representation into irreducible pieces is the goal of Clebsch–Gordan theory. Let be the -dimensional vector space spanned by the states and the -dimensional vector space spanned by the states The tensor product of these spaces, , has a -dimensional uncoupled basis Angular momentum operators are defined to act on states in in the following manner: and where denotes the identity operator. The total angular momentum operators are defined by the coproduct (or tensor product) of the two representations acting on , The total angular momentum operators can be shown to satisfy the very same commutation relations, where . Indeed, the preceding construction is the standard method for constructing an action of a Lie algebra on a tensor product representation. Hence, a set of coupled eigenstates exist for the total angular