# Wavefunction properties tunneling effect

• I
• Salmone
In summary: This means that the eigenfunction for this equation is not zero at any point inside the potential barrier.
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}
\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:
Salmone said:
## \psi_E(x)=\begin{cases}
\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:
vanhees71

## 1. What is a wavefunction?

A wavefunction is a mathematical description of a quantum system, which includes all the information about the system's properties such as position, momentum, and energy.

## 2. What is the tunneling effect?

The tunneling effect is a phenomenon in which a particle can pass through a potential barrier even though it does not have enough energy to overcome the barrier. This is possible due to the probabilistic nature of quantum mechanics.

## 3. How does the wavefunction affect the tunneling effect?

The wavefunction of a particle determines its probability of tunneling through a potential barrier. A particle with a higher wavefunction amplitude has a higher probability of tunneling compared to a particle with a lower wavefunction amplitude.

## 4. What are some properties of wavefunctions in relation to the tunneling effect?

The properties of wavefunctions that affect the tunneling effect include the amplitude, phase, and shape of the wavefunction. A higher amplitude and a longer wavelength of the wavefunction increase the probability of tunneling.

## 5. How does the tunneling effect impact real-world applications?

The tunneling effect has important implications in various fields such as electronics, where it is used in the development of tunneling diodes and transistors. It also plays a crucial role in nuclear fusion reactions and scanning tunneling microscopy used in nanotechnology.

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