Triplet States and Wave Functions

[sungholee]

Why is the triplet state space wave function Ψ_T1=[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?

 

[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.

 

[sungholee]

Thanks for the reply.

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?

 

[blue_leaf77]

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?

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.

 

[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.

 

[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?

 

[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?

 

[blue_leaf77]

only the antisymmetric space wave function can be the triplet for a hydrogen molecule.

Yes, only antisymmetric spatial wavefunction can be paired with the triplet spin state.

As opposed to the space wave function itself having a singlet or triplet characteristic.

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.

 

[sungholee]

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

I still don't see why you are not allowed to have symmetric spatial paired with a singlet spin state

?

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?

 

[blue_leaf77]

what exactly does [α(1)β(2)-β(1)α(2)] imply?

That means the two particles cannot be in the same state, if you force $\alpha = \beta$, the wavefunction will vanish.

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?

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.