# Difference between angular momentum $L$ and $J$ in quantum mechanics

what is the difference between angular momentum L and generalized angular momentum J and their components?

The total angular momentum $$\mathbf{J}$$ is the sum of the orbital angular momentum $$\mathbf{L}$$ and the spin angular momentum $$\mathbf{S}$$.

Spin is an intrinsic type of angular momentum that elementary particles can have even when they are not moving. It is quantum mechanical, not classical, in nature. Despite the use of the word “spin”, the particles are not actually spinning around a rotation axis. If they were, their magnetic moments would be different than they are.

Orbital angular momentum can be arbitrarily large. It is characterized by an integral quantum number $$\ell$$ which can range from 0 to infinity. Spin angular momentum is determined by the type of particle and is characterized by an integral-or-half-integral quantum number $$s$$. For an electron, $$s$$ is 1/2; for a photon, $$s$$ is 1.

When you have both kinds of angular momentum, the total angular momentum is characterized by a quantum number $$j$$ which ranges from $$|\ell-s|$$ to $$\ell+s$$.

The possible quantum values of $$\mathbf{L}^2$$ are $$\ell(\ell+1)\hbar^2$$. The possible quantum values of $$\mathbf{L}_x$$, $$\mathbf{L}_y$$, and $$\mathbf{L}_z$$ are $$m\hbar$$, where $$m$$ ranges from $$-\ell$$ to $$\ell$$ by 1.

The possible quantum values of $$\mathbf{S}^2$$ are $$s(s+1)\hbar^2$$. The possible quantum values of $$\mathbf{S}_x$$, $$\mathbf{S}_y$$, and $$\mathbf{S}_z$$ are $$m\hbar$$, where $$m$$ ranges from $$-s$$ to $$s$$ by 1.

The possible quantum values of $$\mathbf{J}^2$$ are $$j(j+1)\hbar^2$$. The possible quantum values of $$\mathbf{J}_x$$, $$\mathbf{J}_y$$, and $$\mathbf{J}_z$$ are $$m\hbar$$, where $$m$$ ranges from $$-j$$ to $$j$$ by 1.

You cannot have a quantum state in which all three components of (any kind of) angular momentum are well-defined. However, you can have a state where the square and one component are well-defined. By convention, we call that component the $$z$$-component.