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When it comes to deciding which terms exist for a given configuration I have often seen an argument along the following lines:

For the configuration $np^2$ with $S=1$ it is impossible to form a state with $M_L=2$ without violation of the Pauli-exclusion principle. This then implies that the term $^3D$ does not exist for the state configuration $np^2$.

I have am looking for an explanation why the latter sentence must follow the former. Since it is possible to have a wave function with $L=2$ but $M_L=1$ (say) which this statement does not seem to forbid. It is apparently (see e.g. Foot 2005, p 95) likewise possible to deduce that the terms $^1P$ and $^3S$ do not exist via a similar method. I am confused about how we can deduce the existence of a term simply by looking at the allowed values of $M_S$ and $M_L$. Please can someone explain?

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The point here is that existence works both ways. If the state $L=2$ $M_l=1$ exists, so should the rest ($M_l=2,0,-1,-2$). If you can't make one of them (usually the maximum projection) then the state can't exist. Look up M-scheme in nuclear physics for more information.

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