# Acting on a current density operator in second quantization

I have a current density operator in second quantization in the form:

$\hat{J} = -i\left(\psi^\dagger \partial_x \psi - (\partial_x \psi^\dagger) \psi \right)$

operating on some state:

$|\phi \rangle = \int dy (u(y) \psi(y) + v(y)\psi^\dagger (y))|BCS\rangle$

i.e its some excitation of the BCS state (not in momentum basis, the system has no translational invariance so going into fourier space doesn't really help much).

I would like to calculate $\hat{J} |\phi\rangle$ = ? but i have some problems when i try to simplify the expression, and i get derivatives of delta functions.

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Perhaps it would help if you tell us how you obtained the wave functions, and what is your $\left|\text{BCS}\right\rangle$ state. Or eventually more details of the calculation you've done, such that we can understand where the $\delta$-distribution are coming from. Thanks in advance. –  FraSchelle Dec 20 '13 at 13:54
Well I haven't really obtained the wave function, this is a general expression for an excited state in a superconductor. The BCS state is the ground state of the system, and i have no idea how to represent it in coordinate space. I don't think it really matters anyway, because instead of working with the state you can just work with the operator: $\gamma = \int dx(u\psi + v\psi^\dagger)$. Thanks! –  Koby Yavilberg Dec 21 '13 at 8:27