Exchange Interaction - Helium

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Looking thru some notes one this, I came across something which I cant get my head around.

The Hamiltonian of the system (Helium atom) is H = H_1 + H_2 + W, the energies (kinetic and electron-nucleus) of electron 1, electron 2 and the interaction between the electrons themselves.

To find energy, one does the usual expectation value, whereby the H acts on a ket state, and this is then acted on by a bra state. What gets me here tho is that the explanation uses

1) for the ket, a linear combo of the 2 possible statefunctions, ie. either electron could be considered to be in a particular place.

2) for the bra, he first uses only one state, and then does the whole derivation over again with the other.

I am asking, is this simply the same as uses the linear combo for the bra also???
 

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