I Wavefunction properties tunneling effect

Salmone
Messages
101
Reaction score
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.
 
Last edited:
Physics news on Phys.org
Salmone said:
## \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.
 
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 said:
@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:

Salmone said:
"since the eigenfunctions of SE equation must not be equal to zero in a point with their first derivatives, then ##C \neq 0##".
 
Last edited:
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
Back
Top