Solving Hydrogen Molecule Basis Functions

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The discussion revolves around formulating basis functions for a hydrogen molecule with two indistinguishable electrons in distinct atomic orbitals. The initial solution provided four basis functions corresponding to the spin states, but it was pointed out that six basis functions are needed, including configurations for both AA and BB pairs. The Pauli exclusion principle was clarified, emphasizing that it only prohibits two indistinguishable particles from occupying the same quantum state. The participants acknowledged the need to consider all possible pairings of the electrons in the different atomic states. Ultimately, the conversation highlights the importance of correctly applying quantum principles to derive the complete set of basis functions.
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A molecule of hydrogen. We assume that the only possible states of
its two electrons (indistinguishable) are |A\uparrow\rangle,<br /> |A\downarrow\rangle, |B\uparrow\rangle and |B\downarrow\rangle
(A=1s-orbital at atom A, B=1s-orbital at atom B).
Formulate the 6 basis functions using the four possible single
particle states above. (Don't forget the Pauli-principle!)


Here is my solution:

Total spin S=1 or 0.
|1,1\rangle=|A\uparrow\rangle|B\uparrow\rangle,
|1,0\rangle=\frac{1}{\sqrt{2}}(|A\uparrow\rangle|B\downarrow\rangle+|A\downarrow\rangle|B\uparrow\rangle),
|1,-1\rangle=|A\downarrow\rangle|B\downarrow\rangle,
|0,0\rangle=\frac{1}{\sqrt{2}}(|A\uparrow\rangle|B\downarrow\rangle-|A\downarrow\rangle|B\uparrow\rangle).

Are they basis functions? There should be 6 but I got only 4.
 
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You've written down spin triplet and spin singlet states, and those would be all you had if your single particle states were \left|\uparrow\right&gt; and \left|\downarrow\right&gt;, but your states aren't labelled by just the spin - you have labels A and B too. However, all of the states you have written are pairs A and B. What about AA or BB pairs?
 
Hi, Mute. I think there should be no AA or BB pairs. Because at the same time A or B has only one state.
 
Why do you think that? It's not given anywhere in the problem statement that each atom can only have one electron.

Remember that the Pauli exclusion principle only rules out two indist. particles in the *exact* same state...
 
2Tesla and Mute, thanks. I understand now.

|0,0>=|A up>|A down>
|0,0>=|B up>|B down>

Thanks u for ur tips.
 
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