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I'm struggling with the second quantization formalism. I'd like to derive the hamiltonian of a system with non-interacting particles

[tex]\hat{H}=\int dx\,a(x)^\dagger \left[\frac{\hat{P}}{2m}+V(x)\right]a(x),[/tex]

where [tex]a(x) = \hat{\Psi}(x)[/tex].

I know the second quantized representation of a single-particle operator [tex]\hat{O}[/tex] which is diagonal in the basis [tex]\{|\alpha\rangle\}[/tex]:

[tex]\hat{O}=\sum_i o_{\alpha_i} a_{\alpha_i}^\dagger a_{\alpha_i}[/tex]

My idea was, as a first step, to derive the expression of the linear momentum operator:

[tex]\hat{P}=\sum_i p_{p_i} a_{p_i}^\dagger a_{p_i} =\sum_i \int dx\, \langle x|p_i\rangle a^\dagger(x) \int dx\, \langle p_i|x\rangle a(x) ??[/tex]

The above is probably wrong (and I don't know how to proceed in order to derive an expression which is of similar form as [tex]\hat{H}[/tex])

Any help is much appreciated.

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# Operator in second quantization

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