Expectation value of a real scalar field in p state

[inkskin]

Hello,

I've been trying to find <p'|φ(x)|p> for a free scalar field. and integral of <p'|φ(x)φ(x)|p> over 3d in doing the space
In writing φ(x) as

In doing the first, I get the creation and annihilation operators acting on |p> giving |p+1> and |p-1> which are different from the bra state |p>
which results in zero.

I'm pretty sure I'm missing something

 

[eljose79]

, but I can't figure out what. Can anyone help?
Hello,

Thank you for your forum post. It seems like you are on the right track, but there are a few things that you may have missed.

First, when dealing with a free scalar field, the creation and annihilation operators should act on the vacuum state |0>, not the state |p>. This is because the state |p> represents a single particle with momentum p, while the vacuum state |0> represents the absence of any particles.

Second, the integral of <p'|φ(x)φ(x)|p> over 3d space should give us the probability amplitude for a particle to be found at position x. This can be calculated using the Fourier transform of the field operator φ(x), which is given by:

φ(x) = ∫ d^3p (2π)^3 √(ω_p) (a_p e^(ip·x) + a_p^† e^(-ip·x))

where a_p and a_p^† are the creation and annihilation operators, and ω_p is the energy of a particle with momentum p.

Finally, when calculating <p'|φ(x)|p>, we need to use the commutation relation between the creation and annihilation operators, which is given by:

[a_p, a_q^†] = (2π)^3 δ^(3)(p-q)

This will help you simplify the expression and get a non-zero result.

I hope this helps. Let me know if you have any further questions. Good luck with your research!