Sunday, September 17, 2017

wavefunction - Is the Singlet state for Helium with 2 electrons symmetric rather than anti-symmetric as is meant to be for fermions?



I'm looking at two-electron Helium atoms where one electron is in the ground state (due to if it were in other states, it's de-excitation would simply lead to the ionization of the electron). The other electron is in an arbitrary excited state. What does this notation mean in terms of the allowed wavefunctions?: enter image description here


Specifically, for the singlet state, I understand that the both electrons must have opposite spin to give a total overall spin of Zero (correct me if I'm wrong). But why does this mean suddenly the wavefunction, ψ+ is allowed to be symmetric. I thought all fermion wavefunctions had to be anti-symmetric havinga ψ state.


For the triplet state, as I understand it, the electrons have a total spin number of 1 meaning that they have to have the same spin as each other (+1/2 each or -1/2 each). Is this correct? Further to this, I don't specifically understand the notation in the equations above. Specfically, I don't understand the last bit added on the end: enter image description here What does this actually mean?


I understand this might be a silly question as it could just be me mixing up notation but any help will be appreciated. Thank You! :)


EDIT: See AV23 Comment below. Answer! :) Basically, find the ˆS2 eigenvalues to deduce S using below:


enter image description here



Answer



The |00 and |1,MS represent the spin singlet and triplet states. The overall wavefunction must contain both the 'space' part and the 'spin' part. We can schematically express this as follows: ψψ(r1,r2)|s


Now, Pauli's exclusion principle demands the antisymmetry of the overall wavefunction. For the singlet state, as |00=12(|+|+) is already antisymmetric, ψ+ can be antisymmetric only if the space part is symmetric.


The triplet states: |1,1=|++ |1,0=12(|++|+) |1,1=| are all symmetric, and therefore, the space part corresponding to them must be antisymmetric.



Note that the triplet state |1,0 does not have both individual spins in the same direction. However, it still has a total spin of 1.


In summary, ψ+ and ψ are both antisymmetric; it is just that their space & spin parts are symmetric & antisymmetric and antisymmetric & symmetric respectively.


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