I How do you normalize this wave function?

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The discussion centers on normalizing wave functions for a Hamiltonian with a Dirac delta potential, specifically addressing the eigenstates' orthogonality and normalization. The eigenfunctions exhibit both odd and even parity, with the even-parity states including a bound state and extended states forming a continuum. Participants highlight that while these wave functions cannot be normalized in the traditional sense, they can be treated using Dirac delta normalization, which is essential for scattering states. The conversation also touches on the need for renormalization of singular Hamiltonians, particularly in one-dimensional cases, and the challenges of proving orthogonality in the continuous spectrum. Ultimately, the normalization and orthogonality of these wave functions remain complex topics requiring careful mathematical treatment.
  • #31
Isaac0427 said:
Griffiths states for an integral like that (specifically with showing that the integral of exp(ikx) from negative infinity to infinity is zero), although it does not converge, you can replace infinity with L (which I see you have done), and take the average value of the integral as L goes to infinity. This takes you to zero. Thus (based on Griffith’s text— and he acknowledges that it would drive a mathematician insane) I would conclude that the previously mentioned sine integral would be equal to zero when k=k’ (which, of course, is the result we want).

Really many thanks. But I don’t think that strategy works here. That average would not yield zero for the integral.
 
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