# Plotting Quantum Tunneling the real way

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

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.

• What makes you think that $k_2$ must be real? Mar 8, 2022 at 15:44
• 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.
– Leon
Mar 10, 2022 at 10:23
• 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 Mar 10, 2022 at 16:40

## 1 Answer

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.

• 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.
– Leon
Mar 10, 2022 at 10:26