- #1
indigojoker
- 246
- 0
I need to find the number variance [tex]\langle (\Delta N ) ^2 \rangle [/tex] for the state [tex]| \beta \rangle = e^{\alpha a^{\dagger}-\alpha^{*} a}|1 \rangle [/tex]
we know:
[tex]\langle (\Delta N ) ^2 \rangle [/tex]
[tex]\langle a^{\dagger} a a^{\dagger} a \rangle [/tex]
[tex]\langle a^{\dagger} (a^{\dagger}a +1) a \rangle [/tex]
[tex]\langle a^{\dagger} a^{\dagger}a a+a^{\dagger} a \rangle [/tex]
[tex]\langle \beta| a^{\dagger} (a^{\dagger}a +1) a |\beta \rangle [/tex]
I know the relation (since this was derived):
[tex][a^{\dagger},e^{\alpha a}]=-\alpha e^{\alpha a} [/tex]
[tex][a,e^{\alpha a^{\dagger}}]=\alpha e^{\alpha a^{\dagger}} [/tex]
I could expand:
[tex]\langle 1|e^{\alpha a^{\dagger}-\alpha^{*} a} a^{\dagger} (a^{\dagger}a +1) a e^{\alpha a^{\dagger}-\alpha^{*} a}|1 \rangle [/tex]
But I'm not sure how to apply the relation. any ideas would be appreciated.
we know:
[tex]\langle (\Delta N ) ^2 \rangle [/tex]
[tex]\langle a^{\dagger} a a^{\dagger} a \rangle [/tex]
[tex]\langle a^{\dagger} (a^{\dagger}a +1) a \rangle [/tex]
[tex]\langle a^{\dagger} a^{\dagger}a a+a^{\dagger} a \rangle [/tex]
[tex]\langle \beta| a^{\dagger} (a^{\dagger}a +1) a |\beta \rangle [/tex]
I know the relation (since this was derived):
[tex][a^{\dagger},e^{\alpha a}]=-\alpha e^{\alpha a} [/tex]
[tex][a,e^{\alpha a^{\dagger}}]=\alpha e^{\alpha a^{\dagger}} [/tex]
I could expand:
[tex]\langle 1|e^{\alpha a^{\dagger}-\alpha^{*} a} a^{\dagger} (a^{\dagger}a +1) a e^{\alpha a^{\dagger}-\alpha^{*} a}|1 \rangle [/tex]
But I'm not sure how to apply the relation. any ideas would be appreciated.