Wavefunctions of Singlet and Triplet States

[fayled]

If we have a system of two electrons, addition of angular momentum tells us that the spin states of the composite system can be decomposed into those of the two electrons as follows
|1,1>=|+>|+>
|1,0>=(|+>|-> + |->|+>)√2
|1,-1>=|->|->
|0,0>=(|+>|-> - |->|+>)√2
where the states are |s,m_s> for the composite system and |±> is the spin up or down state of an electron.

From this, if you switch the electrons around, the first three (triplet) states are unchanged (symmetric), whilst the final (singlet) state picks up a sign (antisymmetric).

My book states that to fully specify a wavefunction of two electrons, we need to specify four wavefunctions of the form Ψ₊₊(x,x'), Ψ_-+(x,x'), Ψ_+-(x,x') and Ψ_--(x,x') which give the amplitudes of finding the first electron at x and the second at x' with spin states denoted by the subscripts, which it then writes as a vector: <x,x'|Ψ>=(Ψ₊₊,Ψ_-+,Ψ_+-,Ψ_--). By exchange symmetry, all four of these wavefunctions pick up a sign when you swap the two electrons.

If we have the state |Ψ>=|1,1> then clearly the amplitude of measuring the two electrons anywhere such that the spin states aren't ++ is zero, so the vector reduces to <x,x'|Ψ>=(Ψ₊₊,0,0,0). Likewise for |Ψ>=|->|->.

However for the remaining states, things get a bit weird in two ways. Looking at the remaining triplet state, we can deduce that the vector becomes <x,x'|Ψ>=(0,Ψ_-+,Ψ_+-,0), but for some reason my books claims that the two non-vanishing wavefunctions are actually identical. Why is this?

Looking at the singlet state, we can deduce the vector becomes <x,x'|Ψ>=(0,Ψ_-+,Ψ_+-,0) as before, but now my book claims that the two non-vanishing wavefunctions differ only in sign. Again, why is this so?

There is a further issue with the singlet state - writing the vector as <x,x'|Ψ>=Ψ₀(x,x')(0,1,-1,0), my book shows that Ψ₀ is symmetric under a swap of its arguments, yet clearly Ψ₀=Ψ_-+ which is anti-symmetric under a swap of its arguments, so how does that work? Also, this means that all the triplet wavefunctions are antisymmetric and the singlet wavefunction is symmetric, in contrast to what I deduced for the states at the beginning of this post - why is that.

Thanks for any answers!

 

[The_Duck]

The fundamental requirement is that the state of two identical fermions must be antisymmetric under exchange of the two particles. This means exchanging both their positions and their spins. This means that

$\psi_{+-}(x, y) = - \psi_{-+}(y, x)$ (note that both spins and positions have been exchanged)

$\psi_{++}(x, y) = - \psi_{++}(y, x)$ (again both spins and positions have been exchanged, though the spins are identical so you can't really tell)

$\psi_{--}(x, y) = -\psi_{--}(y, x)$

Those are the overall requirements on the state.

This has the following consequences: if the spin state is symmetric (a triplet state) then the spatial wave function must be antisymmetric to maintain the overall asymmetry. If the spin state is antisymmetric (the singlet state) then the spatial wave function must be symmetric to maintain the overall asymmetry.

For example let's say the spin state is the triplet state. The triplet state is unchanged (symmetric) if you exchange the spins of the two particles, as you can see from the spin states you wrote out at the beginning. In wave function language this means

$\psi_{+-}(x, y) = \psi_{-+}(x, y)$ (note I haven't swapped the positions in this equation)

If we combine this with the overall antisymmetry requirement $\psi_{+-}(x, y) = - \psi_{-+}(y, x)$, we find

$\psi_{-+}(x, y) = -\psi_{-+}(y, x)$ -- the spatial wave function is antisymmetric. As promised, the symmetric spin state is "compensated" by an antisymmetric spatial wave function to maintain overall antisymmetry of the state.

 

[fayled]

For example let's say the spin state is the triplet state. The triplet state is unchanged (symmetric) if you exchange the spins of the two particles, as you can see from the spin states you wrote out at the beginning. In wave function language this means

$\psi_{+-}(x, y) = \psi_{-+}(x, y)$ (note I haven't swapped the positions in this equation)

Thanks for the reply - I think I'm close to understanding it. I can't quite understand what I've quoted above though - how does this follow (maybe a mathematical explanation will help me better)?

Edit: I can't see how what you've said holds mathematically - i.e in the triplet state, write the overall wavefunction as a superposition of the two other wavefunctions,
Ψ=(Ψ_+-+Ψ_-+)/√2
now swapping spins just swaps the subscripts in each wavefunction, leaving the same overall wavefunction - I don't see how this is a condition for the two wavefunctions to equal.

 

[fayled]

Can anybody answer my follow up question? Thanks :)

 

[The_Duck]

Edit: I can't see how what you've said holds mathematically - i.e in the triplet state, write the overall wavefunction as a superposition of the two other wavefunctions,
Ψ=(Ψ_+-+Ψ_-+)/√2
now swapping spins just swaps the subscripts in each wavefunction, leaving the same overall wavefunction - I don't see how this is a condition for the two wavefunctions to equal.

The total wave function isn't a sum $\psi_{++} + \psi_{+-} + \psi_{-+} + \psi_{--}$. It's a 4-component vector $(\psi_{++}, \psi_{+-}, \psi_{-+}, \psi_{--})$. Swapping the two spins means to exchange the central two components of the vector. This only leaves the overall vector unchanged if the central two components are equal.