We know that the solutions of time-independent Dirac delta potential well contain bound and scattering states:(adsbygoogle = window.adsbygoogle || []).push({});

$$\psi_b(x)=\frac{\sqrt{mu}}{\hbar}e^{-\frac{mu|x|}{\hbar^2}}\text{ with energy }E_b=-\frac{mu^2}{2\hbar^2}$$

and

$$

\psi_k(x)=

\begin{cases}

A(e^{ikx}+\frac{i\beta}{1-i\beta}e^{-ikx}) \quad &x \leq0\\

\frac{A}{1-i\beta}\quad &x\geq0

\end{cases}

\text{ with energy }E_k=\frac{\hbar^2k^2}{2m}.

$$

My question is how to construct a normalizable time-dependent general solution like in free particle case

$$

\Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty\phi(k) e^{ikx-\frac{\hbar k^2t}{mt}}\mathrm d k,

$$

where

$$\phi(k)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty\Psi(x,0)e^{-ikx}\mathrm dx?$$

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# I General Solution of Dirac Delta Potential Well

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