I'm trying to prove that the position-space representation of a single particle wave function of the state |p⟩ in Quantum Field theory is given by eip⋅x
i.e. ⟨0|ϕ(x,t)|p⟩=eip⋅x, where |p⟩=√2Epa†p|0⟩ is the normalised single particle state.
Here is my attempt to prove it. ⟨0|ϕ(x,t)|p⟩=⟨0|∫d3p′(2π)31√2Ep′(ap′eip′⋅x+a†p′e−ip′⋅x)√2Epa†p|0⟩=∫d3p′(2π)3√EpEp′(⟨0|ap′a†p|0⟩eip′⋅x+⟨0|a†p′a†p|0⟩e−ip′⋅x)
For the second term I'm not as confident to what it should equal to? I'm trying to use the following trick and write ⟨0|a†p′a†p|0⟩ as (ap′|0⟩)†a†p|0⟩ and use the fact that ap′|0⟩=0, but I don't know if this is the proper way to do it.
If ⟨0|a†p′a†p|0⟩ is zero I arrive at the desired result.
⟨0|ϕ(x,t)|p⟩=∫d3p′(2π)3√EpEp′(2π)3δ(3)(p′−p)eip′⋅x=eip⋅x
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