# Klein-Gordon & some potential

1. Nov 26, 2012

### Sekonda

Hello,

I am confused with the use of the potential due to some scalar particle interaction and it's substitution into the Klein-Gordon equation. The KG equation & potential is:

$$(\frac{\partial^2 }{\partial t^2}-\bigtriangledown^2+m^2)\Psi=\delta V\Psi\: ,\: \delta V=\lambda\Psi_{f'}^{*} \Psi_{i'}$$

where the Psi with no sub/superscript is just the mediator in this scalar interaction (the propagator?). Upon substitution we obtain:

$$(\frac{\partial^2 }{\partial t^2}-\bigtriangledown^2+m^2)\Psi=\lambda\Psi_{f'}^{*} \Psi_{i'}\Psi$$

With the solution that:

$$-\frac{\lambda\Psi_{f'}^{*}\Psi_{i'}}{E^2-\mathbf{p}^2-m^2}$$

Where E and p are the differences in the energies and momenta of the i' and f' states, though this solution only works if there is no Psi without any sub/superscripts on the RHS of the KG equation. I think I may be using the potential incorrectly?

Thanks,
SK

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