Wavefunction properties tunneling effect

[Salmone]

I am considering tunnel effect with a potential barrier of a certain height that is $\neq 0$ only for $0 \le x \le a$ . I write the Hamiltonian eigenfunctions outside the barrier as:$\psi_E(x)=\begin{cases} e^{ikx}+Ae^{-ikx} \quad \quad x \le0 \\ Ce^{ikx} \quad \quad x\ge a \\ \end{cases}$
where $k^2=\frac{2mE}{\hbar^2}$. This system represents a particle that goes from $\infnty$ to $0$, one part crosses the potential barrier and continues and one part goes back.

Now what I read in my notes is

"since the eigenfunctions of SE equation must not be equal to zero in a point with their first derivatives, then $C \neq 0$".

How can I prove this statement? I think it is related to Cauchy's problem but I don't know how this implies that the eigenfunction would be equal to zero everywhere.

 

[PeroK]

$\psi_E(x)=\begin{cases} e^{ikx}+Ae^{-ikx} \quad \quad x \le0 \\ Ce^{ikx} \quad \quad x\ge a \\ \end{cases}$
where $k^2=\frac{2mE}{\hbar^2}$.

Now what I read in my notes is

"since the eigenfunctions of SE equation must not be equal to zero in a point with their first derivatives, then $C \neq 0$".

If $C = 0$, then the eigenfunction is identically zero for $x \ge a$. I assume there are physical considerations that do not allow that.

 

[vanhees71]

We cannot answer your question, because you don't describe the specific setup considered. In QT you have to be very precise in the problem statement. Otherwise there's no chance to understand anything. Obviously your wave function is not defined in the interval $(0,a)$. So even your state is not completely defined.

 

[Salmone]

@PeroK @vanhees71 I've edited the question.

 

[PeroK]

@PeroK @vanhees71 I've edited the question.

It's been a while since I've looked at these problems, but I thought the coefficients on either side of the barrier were determined by the continuity of $\psi(x)$ and $\frac{\partial \psi}{\partial x}$ at the boundary of the barrier. So, you would need also to consider the wavefunction in the region $0 < x < a$. That would force $C \ne 0$.

I don't understand what this means:

"since the eigenfunctions of SE equation must not be equal to zero in a point with their first derivatives, then $C \neq 0$".