In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Such an operator is applied to a mathematical representation of the physical state of a system and yields an angular momentum value if the state has a definite value for it. In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion. There are several angular momentum operators: total angular momentum (usually denoted J), orbital angular momentum (usually denoted L), and spin angular momentum (spin for short, usually denoted S). The term angular momentum operator can (confusingly) refer to either the total or the orbital angular momentum. Total angular momentum is always conserved, see Noether's theorem. Overview In quantum mechanics, angular momentum can refer to one of three different, but related things. Orbital angular momentum The classical definition of angular momentum is . The quantum-mechanical counterparts of these objects share the same relationship: where r is the quantum position operator, p is the quantum momentum operator, × is cross product, and L is the orbital angular momentum operator. L (just like p and r) is a vector operator (a vector whose components are operators), i.e. where Lx, Ly, Lz are three different quantum-mechanical operators. In the special case of a single particle with no electric charge and no spin, the orbital angular momentum operator can be written in the position basis as: where is the vector differential operator, del. Spin angular momentum There is another type of angular momentum, called spin angular momentum (more often shortened to spin), represented by the spin operator . Spin is often depicted as a particle literally spinning around an axis, but this is only a metaphor: spin is an intrinsic property of a particle, unrelated to any sort of (yet experimentally observable) motion in space. All elementary particles have a characteristic spin, which is usually nonzero. For example, electrons always have "spin 1/2" while photons always have "spin 1" (details below). Total angular momentum Finally, there is total angular momentum , which combines both the spin and orbital angular momentum of a particle or system: Conservation of angular momentum states that J for a closed system, or J for the