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_{1}+ l

_{2}) will be symmetric under the interchange of the two particles, (L = l

_{1}+ l

_{2}- 1) would be anti-symmetric....and the symmetry under exchange will alternate until we reach the states with (L = l

_{1}- l

_{2}).

If this is all correct, then in order to keep the overall state for two electrons fermionic under LS coupling, we can only combine symmetric total orbital angular momentum states with anti-symmetric total spin angular momentum states (and vice-versa). This page pretty much sums up what I'm trying to say http://quantummechanics.ucsd.edu/ph130a/130_notes/node322.html

But this would mean we can't have states such as:

3s3p

^{3}P

_{1}as L = 1 (symmetric) and S = 1 (also symmetric)

However, my lecture notes use this state as an example to show the effect of residual electrostatic Hamiltonian splitting. So what's wrong?