1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook