pellman
Dec23-09, 03:35 PM
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 a|\zeta\rangle=\zeta |\zeta\rangle. \zeta is a Grassmann (anti-commuting) number.
Show that
a^\dag|\zeta\rangle=-\frac{\partial}{\partial\zeta}|\zeta\rangle.
2. Relevant equations
\{a,a^\dag\}=aa^\dag + a^\dag a = 1
\{a^\dag\,a^\dag\}=0
\{a,a\}=0
3. The attempt at a solution
For small \Delta\zeta we have
|\zeta+\Delta\zeta\rangle = |\zeta\rangle+\Delta\zeta\frac{\partial}{\partial\ zeta}|\zeta\rangle
(Actually this is exact since \Delta\zeta^2=0.)
so we could show first that
|\zeta+\Delta\zeta\rangle = |\zeta\rangle-\Delta\zeta a^\dag|\zeta\rangle
That is as far as I have gotten. And it is not clear to me.. do the anti-commuting c-numbers \zeta commute with the operator a^\dag?
Warren Siegel, Fields, ex. IA2.3(b)
Define the eigenstate of the fermionic annihilation operator as a|\zeta\rangle=\zeta |\zeta\rangle. \zeta is a Grassmann (anti-commuting) number.
Show that
a^\dag|\zeta\rangle=-\frac{\partial}{\partial\zeta}|\zeta\rangle.
2. Relevant equations
\{a,a^\dag\}=aa^\dag + a^\dag a = 1
\{a^\dag\,a^\dag\}=0
\{a,a\}=0
3. The attempt at a solution
For small \Delta\zeta we have
|\zeta+\Delta\zeta\rangle = |\zeta\rangle+\Delta\zeta\frac{\partial}{\partial\ zeta}|\zeta\rangle
(Actually this is exact since \Delta\zeta^2=0.)
so we could show first that
|\zeta+\Delta\zeta\rangle = |\zeta\rangle-\Delta\zeta a^\dag|\zeta\rangle
That is as far as I have gotten. And it is not clear to me.. do the anti-commuting c-numbers \zeta commute with the operator a^\dag?