Arturo Miranda
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I've been having troubles resolving the Schödinger's time independent one-dimensional equation when you have a particle that goes from a zone with a constant potential to a zone with another constant potential, yet the potential is a continuos function of the form:
$$
V(x)=\left\{
\begin{array}{lcl}
0&\text{if}&x<0\\
\displaystyle\frac{V_{0}}{d}x&\text{if}&0<x<d\\
V_{0}&\text{if}&d<x
\end{array}\right.
$$
My main problem is around the solution in the second region of the potential, the non-constant region, in which looks like:
$$E\psi(x)=\frac{\hbar^{2}}{2m}d_{x}^{2}\psi(x)+\frac{V_{0}}{d}x\,\psi(x)$$
If tried solving the differential equation by lowering it's order, yet I have not managed to do so. Is there another way of attacking the problem? Or how may I resolve the diff. equation?
$$
V(x)=\left\{
\begin{array}{lcl}
0&\text{if}&x<0\\
\displaystyle\frac{V_{0}}{d}x&\text{if}&0<x<d\\
V_{0}&\text{if}&d<x
\end{array}\right.
$$
My main problem is around the solution in the second region of the potential, the non-constant region, in which looks like:
$$E\psi(x)=\frac{\hbar^{2}}{2m}d_{x}^{2}\psi(x)+\frac{V_{0}}{d}x\,\psi(x)$$
If tried solving the differential equation by lowering it's order, yet I have not managed to do so. Is there another way of attacking the problem? Or how may I resolve the diff. equation?