- #1

Salmone

- 101

- 13

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.

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.

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