Angular Momentum Operator - Spin, Orbital, and Total Angular Momentum

Spin, Orbital, and Total Angular Momentum

The classical definition of angular momentum is . This can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator. L is then an operator, specifically called the orbital angular momentum operator. Specifically, L is a vector operator, meaning, where L_x, L_y, L_z are three different operators.

However, there is another type of angular momentum, called spin angular momentum (more often shortened to spin), represented by the spin operator S. Almost all elementary particles have spin. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: Spin is an intrinsic property of a particle, unrelated to any sort of motion in space. All elementary particles have a characteristic spin, for example electrons always have "spin 1/2" while photons always have "spin 1".

Finally, there is total angular momentum J, 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 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.