Friday, 16 October 2020

homework and exercises - Prove that the position-space representation of a single particle wave function is given by eitextbfpcdottextbfx


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 eipx


i.e. 0|ϕ(x,t)|p=eipx, where |p=2Epap|0 is the normalised single particle state.


Here is my attempt to prove it. 0|ϕ(x,t)|p=0|d3p(2π)312Ep(apeipx+apeipx)2Epap|0=d3p(2π)3EpEp(0|apap|0eipx+0|apap|0eipx)

Focusing on 0|apap|0 and 0|apap|0, I get that the first term is equal to 0|apap|0=p|p=(2π)3δ(3)(pp)


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|apap|0 as (ap|0)ap|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|apap|0 is zero I arrive at the desired result.


0|ϕ(x,t)|p=d3p(2π)3EpEp(2π)3δ(3)(pp)eipx=eipx

Basically I need someone to confirm if my proof of 0|apap|0=0 is correct or if there's any other property that makes 0|apap|0 zero.




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