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Suppose I have a 2 state system s.t. $H\psi_1=E_1\psi_1$ and $H\psi_2=E_2\psi_2$.

And I have $\phi_+={1\over \sqrt2}(\psi_1+\psi_2)$ and $\phi_-={1\over \sqrt2}(\psi_1-\psi_2)$

Would I be right to think then that $\langle H\rangle$ for both states + and - are equal to ${1\over 2}(E_1+E_2)$? Isn't this weird?? Thanks.

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It would have been good style to link to your very directly related earlier question at…. The answer is, yes, you would be right to think that; it follows directly from what I wrote in my answer to that other question. No, it isn't weird. I suspect that to get a more useful answer than that you'll have to tell us something about why it seems weird to you. – joriki Oct 24 '11 at 11:25
up vote 2 down vote accepted

There's no need to think it. You can compute it, and you would get that $\langle H \rangle = \frac{E_1 + E_2}{2}$.

Don't know what you mean with weird. It's not weird to me... You should expect the same result for $\left| \phi (\alpha, \beta) \right\rangle = \displaystyle\frac{e^{i\alpha}}{\sqrt{2}} \left( \left| \psi_1 \right\rangle + e^{i\beta} \left| \psi_2 \right\rangle \right)$, for any arbitrary values of $\alpha$ and $\beta$.


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