I'm currently reading Griffith's book on time independent Schroedinger's equation about delta functions.(adsbygoogle = window.adsbygoogle || []).push({});

However, I complete dislike how the book deals with the delta distribution.

firstly, the book discusses how to solve:

[tex]-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}-\alpha\delta\psi=E\psi[/tex]

for E<0, by solving the equation piece-wise, and noticing that psi must be continuous:

[tex]\psi=\sqrt k e^{-k|x|}[/tex]

however, so far the alpha has not come into play, the book incorporate alpha by integrating the Schroedinger's equation from -epsilon to +epsilon, then take the limit of epsilon to zero.

However, in doing so, the first integration becomes (according to the book)

[tex]\left \lim_{\epsilon\rightarrow 0}\frac{d\psi}{dx} \right |_{-\epsilon}^{+\epsilon}[/tex]

that is complete bs to me. psi is not even twice differentiable in the usual sense. The whole idea is psi is solved to be a distribution, and the derivative is a distribution, how in the heck can the book just invoke the fundamental theorem of calculus?? at least if they do that, they should at least realize that the integral is improper (zero is a bigggg discontinuity) and make the integral:

[tex]\left \lim_{\epsilon\rightarrow 0}\frac{d\psi}{dx} \right |_{-\epsilon}^{0}+\left \frac{d\psi}{dx} \right |_{0}^{+\epsilon}

[/tex]

If anybody can provide additional insights and a more rigorous treatment of this thing... I'll be greatly appreciated. I want to understand psi in the distributional sense.

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# Time independent schrodinger equation: delta potential

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