Let's say, I have the following electronic configuration, $1s2s$, and I'm trying to find all the possible states.
By looking at the configuration, it is obvious, that in the $|l_1,m_{l_1},m_{s_1},l_2,m_{l_2},m_{s_2}\rangle$ basis, we have a total of $4$ possible states. Since total spin $(S)$ is either $0$ or $1$, w should have one singlet and one triplet giving us a total of 4 states. There are exactly 4 possible electronic configurations.
$\sum_i (2S_i +1)(2L_i+1) = 4$
However in the basis $|L,S,m_L,m_S\rangle$, we get four possible states, as $m_S = 0(singlet);0,1,-1(triplet)\space\space\space m_L=0$. Hence there are four possible configurations.
Similarly, if we represent this in the coupled basis $|L,S,J,m_j\rangle$, we have the two cases : $L,S=0$ and $L=0,S=1$. In the first case, the only value of $J$ is $0$, and hence the value of $m_j$ is also $0$. In the second case, $L\lt S$, so, the number of possible $J$ values is $(2L+1) = 1$.
Hence we obtain two-term symbol states $^1S_0^1$. Thus we are obtaining only two states, in the coupled basis.
How can we have four possible electronic configurations, or four possible wavefunctions given by slater determinants, but only 2 possible states in the coupled basis ?
Shouldn't the number of term symbols be equal to the no. of possible electronic configurations or the no. of states in the uncoupled basis?
Shouldn't $\sum_i (2J_i +1) = (2L+1)(2S+1)$ be true ? If this is not true, then where are those $2$ states, that are included in the configuration and in the uncoupled basis, but not in the term symbols or coupled basis ?
What am I missing here ? What is the actual no. of states in this system ?
Best Answer
Let's enumerate basis states. Since $l=m_l=0$, I am going to use $n\in(1,2)$. The two spatial wave functions are as $\psi_{n_1,n_2}$ have (anti)symmetric combinations:
$$\psi_S = \frac 1{\sqrt 2}(\psi_{1,2}+\psi_{2,1})$$ $$\psi_A = \frac 1{\sqrt 2}(\psi_{1,2}-\psi_{2,1})$$
The spins have 1 antisymmetric combination:
$$\chi_0^0 = \frac 1{\sqrt 2}(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)$$
and 3 antisymmetric:
$$\chi_1^0 = \frac 1{\sqrt 2}(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle)$$ $$\chi_1^1 = |\uparrow\uparrow\rangle$$ $$\chi_1^{-1} = |\downarrow\downarrow\rangle$$
The allowed states are the totally antisymmetric ones:
$$ \psi_S\chi_0^0$$ $$ \psi_A\chi_1^m\,\,\,m\in(-1,0,1)$$
of which there are 4 (though the total number of states is 8).
Since I symmetrized the spin states, this is already in the coupled basis. Since $L=0$, $J=S$. I think your counting went wrong when you failed to enumerate the spatial wave functions.