General Solution of Dirac Delta Potential Well

[rbwang1225]

We know that the solutions of time-independent Dirac delta potential well contain bound and scattering states:
$$\psi_b(x)=\frac{\sqrt{mu}}{\hbar}e^{-\frac{mu|x|}{\hbar^2}}\text{ with energy }E_b=-\frac{mu^2}{2\hbar^2}$$
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
$$\phi(k)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty\Psi(x,0)e^{-ikx}\mathrm dx?$$

 

[BvU]

By doing the integral ?

 

[rbwang1225]

I suppose the result is
$$\Psi(x,t)=c_b\frac{\sqrt{mu}}{\hbar}e^{-\frac{mu|x|}{\hbar^2}}e^{i\frac{mu^2}{2\hbar^3}t}+
\int\limits_{-\infty}^0\phi(k)(e^{ikx}+\frac{i\beta}{1-i\beta}e^{-ikx})e^{-i\frac{\hbar k^2t}{2m}} \mathrm dk+\int\limits_{0}^\infty\phi(k)\frac{1}{1-i\beta}e^{ikx}e^{-i\frac{\hbar k^2t}{2m}} \mathrm d k,$$
where $$\beta=\frac{mu}{\hbar^2 k}.$$
The question becomes how to find $$c_b$$ and $$\phi(k).$$
I don't know if my general solution is correct of not.
If yes, how to determine the coefficients?

 

[BvU]

@vanhees71 ?

 

[vanhees71]

Looks good. You only have to normalize the scattering states to a $\delta$ distribution,
$$\int_{\mathbb{R}} \mathrm{d} x \psi_k^*(x) \psi_{k'}(x)= \delta(k-k'),$$
and the scattering state should have a factor $\exp(\mathrm{i} k x)$ for $x>0$.

Then the coefficients are given by
$$c_b=\int_{\mathbb{R}} \mathrm{d} x \psi_b^*(x) \psi(x,0), \quad \phi(k) = \int_{\mathbb{R}} \mathrm{d} x \psi_{k}^*(x) \psi(x,0).$$