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In "Quantum mechanics" by Schwabl I found a chapter (1.3.3) about one- and two-particle operators in the second quantization. The derivation was only sketched and contained this equation:

$\sum_{\alpha \neq \beta} \left|i\right\rangle_\alpha \left|j\right\rangle_\beta \left\langle k\right|_\alpha \left\langle l\right|_\beta = \sum_{\alpha, \beta} \left|i\right\rangle_\alpha \left\langle k\right|_\alpha \left|j\right\rangle_\beta \left\langle l\right|_\beta - \sum_\alpha \left|i\right\rangle_\alpha \left\langle k\right|_\alpha \left|j\right\rangle_\alpha \left\langle l\right|_\alpha$

I do not know why the left hand side equals the right. Could you please explain it?

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Two steps have been performed in one. First, the inner ket and bra were swapped; since they act on different spaces $\alpha$ and $\beta$, their order is irrelevant. Then the exclusion $\alpha\ne\beta$ was dropped, and the excluded terms were subtracted out to make up for it.

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