- #1
Livethefire
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Simple Integral: Complex exp --> delta function
My Professor has written this down but I'm having some trouble of precisely where this is coming from:
[tex] \int\psi^*_{f}(\boldsymbol k')\psi_{f}(\boldsymbol k) d^3\boldsymbol r = (2\pi)^3\delta(\boldsymbol k- \boldsymbol k') [/tex]
where
[tex] \psi_{f} = e^{i\boldsymbol k\centerdot \boldsymbol r} [/tex] (Plane wave)
I understand that in general (due to orthogonality) we have:
[tex] < \psi_{m} \vert \psi_{n} > =\delta_{mn} [/tex]
and also that a symmetrical integral about a complex exponential yields a sinc function. Qualitatively in the limit that the boundaries become infinite the sinc function becomes a delta function.
What is confusing me here is where are the pi's coming from?
Is there any explicit way to calculate this? Maybe using a spherical coordinate volume element or is there an easier way?
N.B. This calculation pertains to the calculation of density of states (note the f's on the wavefunctions). I know pi's crop up here and there in those types of equations but I can't seem to piece this together.
Thanks.
My Professor has written this down but I'm having some trouble of precisely where this is coming from:
[tex] \int\psi^*_{f}(\boldsymbol k')\psi_{f}(\boldsymbol k) d^3\boldsymbol r = (2\pi)^3\delta(\boldsymbol k- \boldsymbol k') [/tex]
where
[tex] \psi_{f} = e^{i\boldsymbol k\centerdot \boldsymbol r} [/tex] (Plane wave)
I understand that in general (due to orthogonality) we have:
[tex] < \psi_{m} \vert \psi_{n} > =\delta_{mn} [/tex]
and also that a symmetrical integral about a complex exponential yields a sinc function. Qualitatively in the limit that the boundaries become infinite the sinc function becomes a delta function.
What is confusing me here is where are the pi's coming from?
Is there any explicit way to calculate this? Maybe using a spherical coordinate volume element or is there an easier way?
N.B. This calculation pertains to the calculation of density of states (note the f's on the wavefunctions). I know pi's crop up here and there in those types of equations but I can't seem to piece this together.
Thanks.