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Homework Help: Free-field Dirac Hamiltonian

  1. Jul 10, 2011 #1
    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 = [itex]\int[/itex](d3k/(2\pi)3)[itex]\sum[/itex]s([itex]\widehat{c}[/itex]+s(k)[itex]\widehat{c}[/itex]s(k)+([itex]\widehat{d}[/itex]+s(k)[itex]\widehat{d}[/itex]s(k))

    2. Relevant equations

    <p|q> = 2Ep(2\pi)3[itex]\delta[/itex](3)(p-q)

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

    |-,p,v> = (2Ep)1/2[itex]\widehat{c}[/itex]+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> = [itex]\int[/itex](d3k/{(2\pi)3)[itex]\sum[/itex]s<-,p',v';+,q',r'|[itex]\widehat{c}[/itex]+s(k)[itex]\widehat{c}[/itex]s(k)|-,p,v;+q,r>
    = [itex]\int[/itex](d3k/{(2\pi)3)[itex]\sum[/itex]s<-,p',v'|[itex]\widehat{c}[/itex]+s(k)[itex]\widehat{c}[/itex]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[itex]\delta[/itex](3)(v-v')[itex]\delta[/itex](3)(q-q')

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

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

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

    -- Jonathan
  2. jcsd
  3. Jul 10, 2011 #2
    Nevermind, I found a solution to a similar problem and now I know what I did wrong.
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