# Singlet and Triplet Spin States

1. Feb 7, 2010

### joebo

In a system with two spin 1/2 particles ,
We now ask what are the allowed total spin states generated by adding the spins ${\bf S}= {\bf S}_1 + {\bf S}_2$ ,in fact, they are Singlet and Triplet Spin States

\left( \begin{array}{ll} |1,1\rangle & =\uparrow\uparrow\\ |1,0\rangle & =\frac{1}{\sqrt{2}}(\uparrow\downarrow + \downarrow\uparrow)\\ |1,-1\rangle & =\downarrow\downarrow \end{array} \right)\ s=1\ (\mathrm{triplet})

\left(|0,0\rangle=\frac{1}{\sqrt{2}}(\uparrow\downarrow - \downarrow\uparrow)\right)\ s=0\ (\mathrm{singlet})

so, when s=1 and m=0, many people call it state of parallel spin, but i dont think so , and i also dont understand the state . (or s=1,m=0 ) can you help me? thanks!

2. Feb 7, 2010

### SpectraCat

First of all, please edit your post to put TeX tags around your formatting, as I have done in this repsonse.

Second, I'm not sure I understand your question, is it just about the sematics? "Parallel spin" is a colloquial (and imprecise) way to refer to the s=1 triplet state, because the m=+1 and m=-1 cases correspond to microstates where the two spins have the same projection on a space-fixed axis. As you point out, this is at best a "fuzzy" way to refer to the m=0 state, which is the symmetric linear combination of the microstates where the spins have opposite projections. The s=0, m=0 state corresponds to the anti-symmetric linear combination of the opposite-projection microstates.

3. Feb 8, 2010

### ytuab

The singlet and triplet states are historically important for indicating the existence of "spin".
But if you try to understand the singlet and triplet states as "concrete" things, this may be difficult, I think. (Sorry if I misunderstand you.)
We had better consider these states as "mathematical" systems as SpectraCat shows in #2.

In the case of the triplet state (except the case of $$l = 0$$) , the two electrons of the different orbits have the parallel spin to each other.
So the sum of their spins (of electron 1 and 2) is 1. ($$S =S_{1}+S_{2}= 1$$).
In this state, this $$S$$ has three directions (parallel, antiparallel or perpendicular to the orbital angular momentum($$l$$)).
(Here, the state in which $$S$$ is perpendicular to $$l$$ is a little difficult to imagine.)

But actually the magnetic force between the spin-spin interaction is too small.
So why the two separate electrons can make their spin directions the same?
In QM, this is said to be due to the force caused by the "antisymmetry of the Schrodinger equation". (The Stroy of Spin).

For example, when the $$S$$ is zero, the spin effect vanish, and the normal Zeeman effect is said to be seen.(See this thread)

But actually, even in this state the two electrons of the different orbits are apart from each other. So around the electron 1, the magnetic moment by the electron 1 exist, and the spin-orbital interaction (by the electron 1 itself) can occur. (if you imagine this state concretely).
The spin-orbital interaction means that the spin effect doesn't vanish. This is inconsistent with the fact $$S=0$$.

So only the "mathematical systems" are good to describe these states.

Last edited: Feb 8, 2010
4. Feb 8, 2010

### conway

Thanks, SpectraCat, for fixing the Latex. The triplet states are not hard to visualize because everything they do as a combination of two electrons is no different from what the orbital spin states do with s=1, and in this case we can look at the wave functions of the hydrogen atom for a concrete example.

The +/-1 states have a wave function which goes to zero at the poles and varies around the equator as one cycle of the complex exponential function. The m=0 state has two lobes, positive and negative, in the northern and southern hemispheres. If you play around a little with the geometry and the exponential functions, you can see that the m=0 state is really what you would get as the superposition of plus and minus spin states if you had started out by lining up your states to the x or y axis instead of the z axis.

5. Feb 8, 2010

6. Feb 8, 2010

### conway

I don't remember every discussion I get into on the internet but in fact that one did make a big impression on me. In any event, thank you for not assuming I am totally senile.