# Homework Help: Free-field Dirac Hamiltonian

1. Jul 10, 2011

### jonbones

1. The problem statement, all variables and given/known data

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))

2. Relevant equations

<p|q> = 2Ep(2\pi)3$\delta$(3)(p-q)

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

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

3. The attempt at a solution

<-,p',v';+,q',r'|H|-,p,v;+,q,r> = <-,p',v';+,q',r'|Hc|-,p,v;+q,r>+<-,p',v';+q',r'|Hd|-,p,v;+,q,r>

<-,p',v';+q',r'|Hc|-,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(2Ev)-1<-,p',v'|-,p,v><+q',r'|+q,r>
= (2\pi)32Eq$\delta$(3)(v-v')$\delta$(3)(q-q')

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

I know these are wrong since <-,p',v';+q',r'|Hc|-,p,v;+,q,r> $\propto$Ep and <-,p',v';+q',r'|Hd|-,p,v;+,q,r> $\propto$Eq.

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

2. Jul 10, 2011

### jonbones

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