[QUE/QFT-04009] QFT-PROBLEM

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Using the expansion \[ \psi(x) = \sum a_n u_n(x) \] where \(\{u_n(x)\}\)is a set of orthonormal functions. Assume anti commutation relations for the creation and annihilation operators and

• prove that \[\matrixelement{0}{\psi(x)\psi(y)}{m,n} = \frac{1}{\sqrt{2}}[u_m(x)u_(y)- u_n(x)u_m(y)];\]
• show that \[\matrixelement{m}{\psi(x)\psi(y)}{n} = [u_m^*(x)u_n(y)- u_n(x)u_m^*(y)].\]

The answer in part(a) is just the two particle antisymmetric wave function.