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Can someone explain the attached image for me please?
I do not understand how [tex]2\delta_{k, k'}a_{k'}^{\dagger}a_{k}[/tex] becomes [tex]
a_{k}^{\dagger}a_{k} + a_{k}^{\dagger}a_{k} [/tex] to me it should just be [tex]2a_{k}^{\dagger}a_{k}[/tex]
and also I do not understand how [tex]e^{ik}a_{k}a_{k} + e^{ik}a_{k}^{\dagger}a_{k}^{\dagger} = \cos(k) a_{k}a_{k} + \cos(k) a_{k}^{\dagger}a_{k}^{\dagger}[/tex]
I do not understand how [tex]2\delta_{k, k'}a_{k'}^{\dagger}a_{k}[/tex] becomes [tex]
a_{k}^{\dagger}a_{k} + a_{k}^{\dagger}a_{k} [/tex] to me it should just be [tex]2a_{k}^{\dagger}a_{k}[/tex]
and also I do not understand how [tex]e^{ik}a_{k}a_{k} + e^{ik}a_{k}^{\dagger}a_{k}^{\dagger} = \cos(k) a_{k}a_{k} + \cos(k) a_{k}^{\dagger}a_{k}^{\dagger}[/tex]
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