- #1
rbwang1225
- 118
- 0
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
$$\phi(k)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty\Psi(x,0)e^{-ikx}\mathrm dx?$$
$$\psi_b(x)=\frac{\sqrt{mu}}{\hbar}e^{-\frac{mu|x|}{\hbar^2}}\text{ with energy }E_b=-\frac{mu^2}{2\hbar^2}$$
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
$$\phi(k)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty\Psi(x,0)e^{-ikx}\mathrm dx?$$