Schrodinger equation plays an important role in Quantum Mechanics.
![{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\right]\Psi (\mathbf {r} ,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4894f9a205c471cb012351a1fccdd01285b6f11c)
Despite the mathematical difficulties for analytic solutions, we can always try to obtain the numberical solution. In the following paragraphs, I am going to show the numerical simulation of the tunneling effect and the double-slit inteference based on the time-dependent Schrodinger equation.
The reference is this one.
https://web.pa.msu.edu/people/duxbury/courses/phy480/SchrodingerDynamics.pdf
The python script can be found in my github.
https://github.com/xialigang/schrodinger
Tunneling effect
This effect has been used by Gamow to explain the alpha decay in nuclear physics. It is also the principle behind the scanning-tunneling microscope. It says that a particle can pass through a potential in quantum mechanics, which is forbidden in classic physics.
Let us look at the simulation at different times. We start with a Gaussian wave package. The barrier is put at x=50. Clearly, we can see a part of the wave package penetrating the barrier at late times.
Double-slit interference experiment
The two-slit apparatus is put at x=50. From the second plot, we can clearly see two slits. From the last plot, we can see the interference pattern on both sides of the two-slit barrier. This is because we have scattered (forward) and reflected (backward) waves.
It is interesting absolutely, helping us to see how a wave envolve with time. You are welcome to use the codes in my github ( https://github.com/xialigang/schrodinger ). You only need to install PYTHON and ROOT, which are used in the field of experimental high-energy physics (and other fields ofcourse).
I would say the numerical simulation algorithm is very simple. You can try by youself (one day is enough). The thing I did not tell you is that I spend a lot time adjusting the parameters and spend a lot of CPU time to produce these beautiful plots.
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