Free-field Dirac Hamiltonian

[jonbones]

Homework Statement

This is a simple problem I thought of and I'm get a nonsensical answer.
I'm not sure where I'm going wrong in the calculation.

Find the value of <-,p',v';+,q',r'|H|-,p,v;+,q,r>
where H is the free-field Dirac Hamiltonian

H = $\int$(d³k/(2\pi)³)$\sum$_s($\widehat{c}$⁺_s(k)$\widehat{c}$_s(k)+($\widehat{d}$⁺_s(k)$\widehat{d}$_s(k))

Homework Equations

<p|q> = 2E_p(2\pi)³$\delta$⁽³⁾(p-q)

|+,q,r> = (2E_q)^1/2$\widehat{d}$⁺_r(q)|0>

|-,p,v> = (2E_p)^1/2$\widehat{c}$⁺_v(p)|0>

The Attempt at a Solution

<-,p',v';+,q',r'|H|-,p,v;+,q,r> = <-,p',v';+,q',r'|H_c|-,p,v;+q,r>+<-,p',v';+q',r'|H_d|-,p,v;+,q,r>

<-,p',v';+q',r'|H_c|-,p,v;+,q,r> = $\int$(d³k/{(2\pi)³)$\sum$_s<-,p',v';+,q',r'|$\widehat{c}$⁺_s(k)$\widehat{c}$_s(k)|-,p,v;+q,r>
= $\int$(d³k/{(2\pi)³)$\sum$_s<-,p',v'|$\widehat{c}$⁺_s(k)$\widehat{c}$_s(k)|-,p,v><+,q',r'|+q,r>
= 1/(2\pi)³(2E_v)^-1<-,p',v'|-,p,v><+q',r'|+q,r>
= (2\pi)³2E_q$\delta$⁽³⁾(v-v')$\delta$⁽³⁾(q-q')

Similarly, <-,p',v';+q',r'|H_d|-,p,v;+,q,r> = (2\pi)³2E_v$\delta$⁽³⁾(v-v')$\delta$⁽³⁾(q-q')

I know these are wrong since <-,p',v';+q',r'|H_c|-,p,v;+,q,r> $\propto$E_p and <-,p',v';+q',r'|H_d|-,p,v;+,q,r> $\propto$E_q.

I'm pretty sure I'm calculating the operator pieces like <-,p',v';+,q',r'|$\widehat{c}$⁺_s(k)$\widehat{c}$_s(k)|-,p,v;+q,r> incorrectly but I'm not sure where I'm going wrong.

-- Jonathan

 

[jonbones]

Nevermind, I found a solution to a similar problem and now I know what I did wrong.