Wednesday, December 23, 2015

quantum mechanics - What are the assumptions behind "term symbols"?



In multi-electron atoms, the electronic state of the optically active "subshell" is often expressed in "term symbols" notation. I.e. $^{2S+1}L_J$. This presumes that the system of electrons has definite $L^2$, $S^2$ and $\mathbf{J}$ eigenstates.


In order to determine all the possible electronic states compatible with the Pauli's exclusion principle, the technique is to assign to each electron its $m_s$ and $m_l$ and to see what states can be found in terms of eigenvalues of $L^2$, $S^2$, $J^2$ and $J_z$.


What justifies this procedure? In other words, the assumption so far was:



  • heavy nucleus -> the Hamiltonian is spherically symmetric -> total $\mathbf{J}$ is conserved

  • spin-orbit is negligible to some extent -> total $S^2$ and $L^2$ are conserved


What justifies, apart from intuitiveness, returning to the single electron picture (assigning to every electron its spin and angular momentum) to find what states are allowed or not according to the requirement that the total wavefunction should be antisymmetric? And could be there exceptions?




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