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(adsbygoogle = window.adsbygoogle || []).push({});

|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 atxand the second atx'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>=Ψ_{0}(x,x')(0,1,-1,0), my book shows that Ψ_{0}is symmetric under a swap of its arguments, yet clearly Ψ_{0}=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!

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# Wavefunctions of Singlet and Triplet States

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