I am feeling a little stupid tonight... So let me build the problem...(adsbygoogle = window.adsbygoogle || []).push({});

For a single particle operator [itex]O[/itex], we have in the basis [itex] |i> [/itex] we have that:

[itex] O= \sum_{ij} o_{ij} |i><j| [/itex] with [itex]o_{ij}=<i|O|j>[/itex]

Then for N particles we have that:

[itex] T=\sum_{a}O_{a}= \sum_{ij} o_{ij} \sum_{a} |i>_{a}<j|_{a} [/itex] with [itex]a=1,2...,N[/itex]

How can we write this afterwards (for bosons) in respect to the creation and annihilation operators as:

[itex]T= \sum_{ij} o_{ij} c_{i}^{t} c_{j} [/itex]

From what I suspect, [itex]\sum_{a} |i>_{a}<j|_{a}[/itex] must be equal to the number operator... But for some reason I'm unable to see it...Also I tried taking this:

[itex] O= \sum_{ij} o_{ij} |i><j| [/itex]

[itex] O= \sum_{ij} o_{ij} c_{i}^{t}|0><0|c_{j} [/itex]

And then summing over the particles:

[itex] O= \sum_{ij} o_{ij} c_{i}^{t}c_{j} \sum_{a}|0>_{a}<0|_{a} [/itex]

but I see no reason why the sum must be equal to unity [itex] \sum_{a}|0>_{a}<0|_{a} =1 [/itex]

Please give hints, not answers

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# Second Quantization-Kinetic operator

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