(adsbygoogle = window.adsbygoogle || []).push({}); 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]?

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# Homework Help: Fermionic oscillator property

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