Often when looking at Quantum Tunneling graphs something similar is shown:

enter image description here

Having an exponentially decaying graph inside the barrier. But these graphs seem to be arranged. There I ask myself how to plot a real peace-wise function for quantum tunnelling.

First of all the wave functions for all areas can be written as:

$\mathbf{I}: \quad \psi_1 = A_1\,e^{\textstyle -i\,k_1\,x}+B_1\,e^{\textstyle i\,k_1\,x} \quad \mathbf{II}: \quad \psi_1 = A_2\,e^{\textstyle -i\,k_2\,x}+B_2\,e^{\textstyle i\,k_2\,x} \quad \mathbf{III} \quad \psi_1 = A_3\,e^{\textstyle -i\,k_1\,x} $

After using some boundary conditions ($\psi_1(0) = \psi_2(0) \quad \psi_1(d) = \psi_3(d) $ etc.) the Amplitude $A_3$ for area $\mathbf{III}$ can be formulated: $A_3 = \dfrac{4\,k_1\,k_2\,e^{\textstyle i\,k_1\,d}}{(k_1+k_2)^2\,e^{\textstyle i\,k_2\,d}-(k_1-k_2)^2\,e^{\textstyle -i\,k_2\,d}}$

Which is directly linked to the transmission coefficient $T = |A_3|^2$ approximated fairly by $T \approx 16\,e^{\textstyle-2\,i\,k_2}$

From this point the exponential decay can be intuitively comprehended, but what about plotting the probability amplitude $|\psi_i|^2$ for all waves dependent on travelled distance $x$? Then I do not get the popular graph that I just had set up. In general $|\psi|^2$ seems to be a wave anyhow so never a monotonic decreasing function.

  • 3
    $\begingroup$ What makes you think that $k_2$ must be real? $\endgroup$ Mar 8, 2022 at 15:44
  • $\begingroup$ I should have mentioned that. $k_2$ is defined as $\dfrac{1}{\hbar}\,\sqrt{2\,m\,(E-U)}$. Thus in the approximated case the energy of the wave $E$ has to be smaller than the potential $V$. In the analytic case it's somehow not important whether $k_2$ is real or imaginary. The Transmission coefficient will be real. $\endgroup$
    – Leon
    Mar 10, 2022 at 10:23
  • 1
    $\begingroup$ What is the approximated case and what is the analytic case? Of course it is important! When $k_2$ is imaginary then $\exp(\pm i k_2 x)$ are exponential functions that don't oscillate sinusoidally $\endgroup$ Mar 10, 2022 at 16:40

1 Answer 1


The picture you posted actually looks like a fairly accurate one. I'm not sure what you mean by "arranged". In any case, the wave function-squared $|\psi|^2$ does decrease exponentially and monotonically in region II.

Maybe an animation can help to understand the dynamics; I found one on Wikipedia which shows the tunneling quite nicely.

  • $\begingroup$ Yea, I just sketched the graph by tampering with some functions so it's kind of arranged and not based on physical equations. But now after dealing with the finite square well potential I eventually got the hang of real physics solution that fulfil the boundary conditions and how to plot them. $\endgroup$
    – Leon
    Mar 10, 2022 at 10:26

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