Exchange Interaction - Helium

[Master J]

Looking thru some notes one this, I came across something which I can't 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?

 

[azntoon]

It makes sense to me that you'd need 2 state functions (ket & bra) to get an expectation value, yet I'm not sure this is the same situation?Yes, this is the same as using a linear combination of the two possible state functions for both the ket and the bra. The expectation value of a Hamiltonian is the average energy of a system when it is in a particular state. The expectation value is calculated by taking the inner product of the ket and bra states. This means that both the ket and the bra must be expressed in terms of the same basis set so that the inner product can be calculated correctly. Thus, in this case, a linear combination of the two possible state functions is used for both the ket and the bra.