Not a problem par se, as much as me failing somewhere in algebra, and i cant find my mistakes.(adsbygoogle = window.adsbygoogle || []).push({});

Trying to work through the secular determinant

[tex]

E_\pm = det\left(

\begin{array}{cc}

\alpha - E & \beta - ES\\

\beta - ES & \alpha - E

\end{array}

\right)

= 0

[/tex]

which gives the answer

[tex]

E_\pm = \frac{\alpha \pm \beta} {1 \pm S}

[/tex]

This multiplies out to give

[tex]

(\alpha - E)^2 - (\beta - ES)^2 = 0

[/tex]

which with rearrangement gives

[tex]

(1 - S^2)E^2 + 2(\beta S - \alpha)E + (\alpha^2 - \beta^2) = 0

[/tex]

this is quadratic in E, so solutions will be given by the usual formula for solving quadratic equations. This is where I seem to fail, rather badly really, and I cant find my error. If someone could point me in the direction of somewhere this has been solved fully, each step so i can find my error, id be very much obliged. If not I can write out my own calculations on here, but that is rather time consuming. But I will if needed.

Thanks

Steve

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# Homework Help: Secular determinant in MO method.

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