Help probability amplitud Klein gordon

[arten]

Homework Statement

Hello, I need help. I don't know how to calculate this expression. probability amplitude in the final state of antiparticle with momentum q and particle with momentum p. I should write it in terms of creator and anihilation operators

<p,q|[∫d4x(∅∂i∅*-∅*∂i∅)A(x)+(∅π-∅*π*)B(x)]|0>

Where the functions and momentums ∅,π are the usual ones in the Klein-Gordon free hamiltonian.

Homework Equations

I think it will be useful |p,q>=√2Ep√2Eqa*pb*q|0>

The Attempt at a Solution

Then I expand the spatial derivatives and functions but I get terms like aa*+a*a etc. and I can't write the commutator and I don't know how to proceed

The terms a*a|0>= 0

I don't know which momentum write in each ∅ ∅* etc. in the calculation.

Thanks

 

[mark726]

Thank you for reaching out for help with your calculation. I would be happy to assist you in finding a solution to your problem.

Firstly, I want to clarify that the expression you have provided is the amplitude for a specific process in quantum field theory, where a particle with momentum p and an antiparticle with momentum q annihilate to create a photon with momentum k. This amplitude can be calculated using the Feynman rules for this process, which involve the creation and annihilation operators you mentioned.

To begin, let's break down the expression into smaller parts. The first term, ∫d4x(∅∂i∅*-∅*∂i∅)A(x), represents the interaction between the particle and antiparticle, where A(x) is the electromagnetic field operator. This term can be expanded using the commutation relations between the creation and annihilation operators, but it is important to remember that the momentum p should be assigned to the particle and the momentum q to the antiparticle.

The second term, (∅π-∅*π*)B(x), represents the creation and annihilation of the photon with momentum k. Again, it is important to assign the correct momentum to the creation and annihilation operators.

To proceed with the calculation, you can use the Feynman rules to find the amplitude for this process. I would also recommend consulting a textbook or seeking assistance from a colleague or professor for further guidance.

I hope this helps and good luck with your calculation!