- #1
grilo
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Homework Statement
(from "Advanced Quantum Mechanics", by Franz Schwabl)
Show, by verifying the relation
[tex]\[n(\bold{x})|\phi\rangle = \delta(\bold{x}-\bold{x'})|\phi\rangle\][/tex],
that the state
[tex]\[|\phi\rangle = \psi^\dagger(\bold{x'})|0\rangle\][/tex]
([tex]\[|0\rangle =\][/tex]vacuum state) describes a particle with the position [tex]\bold{x'}[/tex].
Homework Equations
The particle density operator [tex]n(\bold{x})[/tex] is defined as
[tex]n(\bold{x}) = \psi^\dagger(\bold{x})\psi(\bold{x})[/tex]
The Attempt at a Solution
Acting on the given state with the particle density operator, I got
[tex]n(\bold{x})|\phi\rangle = \psi^\dagger(\bold{x})\psi(\bold{x})\psi^\dagger(\bold{x'})|0\rangle = \psi^\dagger(\bold{x})(\delta(\bold{x}-\bold{x'}) \pm \psi^\dagger(\bold{x'})\psi(\bold{x}))|0\rangle[/tex]
by the (fermion) boson (anti-)commutation rules. Since [tex]\psi(\bold{x})[/tex] annihilates the vacuum:
[tex]n(\bold{x})|\phi\rangle = \delta(\bold{x}-\bold{x'})\psi^\dagger(\bold{x})|0\rangle[/tex]
which looks like the given equation, but [tex]\psi^\dagger(\bold{x})|0\rangle[/tex] describes a particle at the position [tex]\bold{x}[/tex].
Integrating the last equation in [tex]\bold{x}[/tex] gives back the original state [tex]|\phi\rangle[/tex], though.
I'm not sure wheter I misunderstood something or it's just a matter of interpretation.
Can anyone help me?