I kind of get the idea of singlet and triplet states. But why are they called singlet and triplet (what is the single and what is the triple in these cases)?
I feel that I am missing something obvious!
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The terms arose back in the early days of quantum physics when spectral lines that were expected to be singlets were actually observed to be more complex (doublets, triplets, etc.).
An electron can have a spin quantum number of $+\frac12$ or $-\frac12$. For a system that exists as a singlet, all spins are paired and the total spin for the system is $S=0$. If we have a single electron, $S=\frac12$. If we have a triplet system with 2 unpaired electrons, $S=1$. Also associated with these electrons/ systems is a spin-angular momentum vector, $L$. Quantum mechanics tells us that $L$ is allowed to have $2S+1$ distinct values. So
for a singlet, $L = 2(0) + 1 = 1$ only one state exists (singlet)
for a doublet, $L = 2\left(\frac12\right) + 1 = 2$ two states exist (doublet)
for a triplet, $L = 2(1) + 1 = 3$ three states exist (triplet)
A pair of electrons, being fermions, must have antisymmetric wave function, i.e. if $\psi(\xi_1,\xi_2)$ is a wavefunction describing the system, where $\xi_1$ are position and spin of electron 1 and $\xi_2$ is position and spin of electron 2, then $\psi(\xi_2,\xi_1)=-\psi(\xi_1,\xi_2)$.
In the first approximation, spin degree of freedom can be separated from orbital degrees of freedom, so that the wavefunction becomes $\chi(s_1,s_2)\phi(x_1,x_2)$, where $s_i$ is spin of $i$th electron, and $x_i$ is position of $i$th electron. Here $\chi$ is spin part of wavefunction, and $\phi$ is orbital part. To preserve total antisymmetry of the wavefunction, $\chi$ and $\phi$ can be either symmetric, or antisymmetric. If one is symmetric, the other must be antisymmetric.
The spin of a single electron can be up $\uparrow$ or down $\downarrow$. I.e. simplest options for a two-electron system could be $\uparrow\uparrow$, $\downarrow\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$. But the latter two don't honour indistinguishability of electrons. To correctly include indistinguishability of electrons, we should take symmetric and antisymmetric linear combinations of these spin states.
Now we have four options, split in two variants:
a) Antisymmetric orbital and symmetric spin part of wavefunction
b) Symmetric orbital and antisymmetric spin part
From here we can see that symmetric spin part of wavefunction gives rise to three different states — these are triplet states. If spin part of wavefunction is antisymmetric, there's only one such state — it's the singlet state.
When one makes spectroscopic measurements with not very high resolution, states with different spins but same orbitals will appear to have the same energies, so the spectral lines will appear the same. But if you put your system in magnetic field, you'll see that the spectral lines split according to spin multiplicities: spin-singlet states will remain single lines, while spin-triplets will split into three different spectral lines. This is the origin of such naming.