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Fermionic oscillator property

  1. Dec 23, 2009 #1
    1. The problem statement, all variables and given/known data
    Warren Siegel, Fields, ex. IA2.3(b)

    Define the eigenstate of the fermionic annihilation operator as [tex]a|\zeta\rangle=\zeta |\zeta\rangle[/tex]. [tex]\zeta[/tex] is a Grassmann (anti-commuting) number.

    Show that

    [tex]a^\dag|\zeta\rangle=-\frac{\partial}{\partial\zeta}|\zeta\rangle[/tex].

    2. Relevant equations

    [tex]\{a,a^\dag\}=aa^\dag + a^\dag a = 1[/tex]
    [tex]\{a^\dag\,a^\dag\}=0[/tex]
    [tex]\{a,a\}=0[/tex]

    3. The attempt at a solution

    For small [tex]\Delta\zeta[/tex] we have

    [tex]|\zeta+\Delta\zeta\rangle = |\zeta\rangle+\Delta\zeta\frac{\partial}{\partial\zeta}|\zeta\rangle[/tex]

    (Actually this is exact since [tex]\Delta\zeta^2=0[/tex].)

    so we could show first that

    [tex]|\zeta+\Delta\zeta\rangle = |\zeta\rangle-\Delta\zeta a^\dag|\zeta\rangle[/tex]

    That is as far as I have gotten. And it is not clear to me.. do the anti-commuting c-numbers [tex]\zeta[/tex] commute with the operator [tex]a^\dag[/tex]?
     
  2. jcsd
  3. Dec 23, 2009 #2
    Actually, I think I about have this one. But it hinges on .. do the anti-commuting c-numbers [tex]\zeta[/tex] commute or anti-commute with the operator [tex]a^\dag[/tex]?
     
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