# I Triplet States and Wave Functions

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1. Mar 29, 2016

### sungholee

Why is the triplet state space wave function ΨT1=[1σ*(r1)1σ(r2)-1σ(r1)1σ*(r2)] (ie. subtractive)? How does it relate to its antisymmetric nature?

Also, why is this opposite for the spin wave function α(1)β(2)+β(1)α(2) (ie. additive)? And why is this one symmetric even though it describes the triplet state?

2. Mar 29, 2016

### blue_leaf77

The total wavefunction including both space and spin degrees of freedom for a fermion must be antisymmetric. If the wavefunction were to be written as a product between the spatial and spin wavefunctions, the preceding statement implies that these two wavefunctions must have opposite symmetry nature. Namely, if the spin wavefunction is symmetric (e.g. triplet states) then the spatial wavefunction must be antisymmetric and vice versa.

3. Mar 29, 2016

### sungholee

I understand that, but I still don't understand why the triplet state for the space is subtractive and for the spin is additive. As in, the product of the two would still be antisymmetric even if the triplet state for the space was additive and for the spin subtractive, but why is that not the case?

4. Mar 29, 2016

### blue_leaf77

I think you should specify which quantum system you are talking about. As you noted, the second possibility with the substractive spin state (such a state is commonly called singlet spin state) is also possible.

5. Mar 29, 2016

### sungholee

For a H2 molecule (for the independent particle model, if that matters).

I guess what I might really be asking then is the physical implication of adding and subtracting the components? (the MOs and the spins) in the wavefunctions.

6. Mar 29, 2016

### sungholee

As in, I understand that the addition leads to symmetric and the subtraction leads to antisymmetric, but how does that relate to the singlet and triplet states?

7. Mar 29, 2016

### sungholee

Actually, I think I just understood it. The single-triplet thing is derived from the spin wave functions and due to fermions having to be antisymmetric overall, only the antisymmetric space wave function can be the triplet for a hydrogen molecule. As opposed to the space wave function itself having a singlet or triplet characteristic. Is that correct?

8. Mar 29, 2016

### blue_leaf77

Yes, only antisymmetric spatial wavefunction can be paired with the triplet spin state.
The triplet-singlet terms are exclusively used for spin states, because it has to do with the manifold the states exhibit regarding their total spin. For spatial wavefunction, using triplet-singlet term is a misuse. Anyway, I still don't see why you are not allowed to have symmetric spatial paired with a singlet spin state. It's equally allowed as that with the triplet spin state, the only difference is the energy.

9. Mar 29, 2016

### sungholee

Thanks, everything makes so much more sense now haha. But what do you mean by
?

Also, final question related to this: what exactly does [α(1)β(2)-β(1)α(2)] imply? I suppose that [α(1)β(2)+β(1)α(2)] means the sum of the two possible spin states (up,down and down,up) which explains the summation but how can we subtract spin states?

10. Mar 29, 2016

### blue_leaf77

That means the two particles cannot be in the same state, if you force $\alpha = \beta$, the wavefunction will vanish.
The coefficients can even be complex. The thing is, a state is described as a vector in Hilbert space and the coefficient of each basis vector is a complex scalar.