How do you normalize this wave function?

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The forum discussion centers on normalizing wave functions in quantum mechanics, specifically for the Hamiltonian $$H = -\frac{\hbar^2}{2m}\partial^2_x - V_0 \delta(x)$$. Participants discuss the eigen wave functions, particularly the extended states represented by $$\psi_k(x) = N_k \left(e^{-ik|x|} + b_k e^{ik|x|}\right)$$, where $$b_k = \frac{iV_0}{2k}+1$$. The normalization factor $$N_k$$ is derived through integration techniques, with emphasis on the orthogonality of wave functions in the continuous spectrum, leading to the conclusion that they can be normalized to the Dirac delta function, albeit with specific considerations regarding the singular nature of the Hamiltonian.

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  • #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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