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I am reading about second quantization. The kinetic energy operatorTwe write as

[tex]

\hat T = \sum\limits_{i,j} {\left\langle i \right|T\left| j \right\rangle } \,c_i^\dag c_j^{}.

[/tex]

Now, the creation and annihilation operators really seem to be analogous (in some sense) to the ket and the bra in first quantization, since they tell us which matrix element we are talking about.

What is the reason for this? I understand that we have the new states in Fock space (the occupation number states), but my book never illuminates why the creation and annihilation operators designate the matrix elements just like the outer product in first quantization does.

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# Second quantization

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