What's the total angular momentum operator for a system of two particles?

[AxiomOfChoice]

Suppose we're in two dimensions, and both particles have mass 1.

Particle 1's location is given by its polar coordinates $(r_1,\theta_1)$; likewise for Particle 2 $(r_2,\theta_2)$.

Is it true that the total angular momentum $\vec{L}$ is just the sum of the individual angular momenta of the particles: $\vec{L} = \vec{L}_1 + \vec{L}_2$? If that's the case, can you give me the total angular momentum operator $\vec{L}$ as a differential operator?

 

[kanato]

yeah, just add it up: $$L_j^z = -i\hbar \left[y_j\partial/\partial{x_j} - {x_j}\partial/\partial{y_j}\right]$$ where j is the particle index. Keep in mind L_2 does not act on the coordinates for the first particle.