Animating Quantum tunneling

I did this for a QM 1 lecture once. I just did the brute-force Fourier decomposition of a wave packet as the one shown in your youtube stream. You just solve the energy-eigenvalue problem (time-independent Schrödinger equation) for your potential well and then Fourier transform the initial wave packet in terms of energy eigenfunction. I just did the one-dimensional integral for each time with a Gauss quadrature. It's not that demanding on CPU time. So you don't need more fancy algorithms for this.

If you like to see the animations, you can find them here (however the text around it is in German):

Free particle:
http://theory.gsi.de/~vanhees/faq/quant/node33.html

total reflection on a finite potential well (step function):
http://theory.gsi.de/~vanhees/faq/quant/node35.html

with a small part running through that well:
http://theory.gsi.de/~vanhees/faq/quant/node35.html

some general case (different particle energies):
http://theory.gsi.de/~vanhees/faq/quant/node37.html
http://theory.gsi.de/~vanhees/faq/quant/node38.html
http://theory.gsi.de/~vanhees/faq/quant/node39.html

Rectangular finite-depth potential pot:

energy close to a resonance value (asymptotic final state is the wave packet nearly completely run through the well but with some time delay)
http://theory.gsi.de/~vanhees/faq/quant/node44.html

and at some other energy:
http://theory.gsi.de/~vanhees/faq/quant/node45.html

wave packet as superposition of some bound states:
http://theory.gsi.de/~vanhees/faq/quant/node46.html

 

You're only supposed to get one probability distribution.
The energy eigenvectors will be these piecewise function for the three regions (left, inside, and right of the barrier), they're supposed to be orthogonal so you can just sum a bunch of them and make a gaussian to the right of the barrier.

 

Of course, the wave packets shown in my animations are just solutions of the time-dependent Schrödinger equation. I'm plotting ##|\psi(t,x)|^2##, i.e., the probability distribution. It's done by numerical integration of the explicit solution in terms of energy eigenfunctions. In the general case you have a discrete spectrum (usually eigenvalues ##E_n<0##) and a continuous one for ##E>0##. Then you take an incoming wave packet far away from the support of the potential or, in the case where the potential is non-zero also at infinity like in the case of the potential step in a region of constant potential ("asymptotic free state") and let it run towards the region, where the potential is active. This produces the scattering solutions shown in the animations. Another posibility is that a particle is trapped in a potential valley as is exemplified for the potential pot. Given the initial wave packet as ##\psi(t=0,x=\psi_0(x)##, the time evolution follows from the expansion in terms of energy eigenfunctions ##u_n## in the discrete and ##u_E## in the continuous spectrum, i.e.,
$$\psi(t,x)=\sum_{n} u_n(x) \psi_n \exp(-\mathrm{i} E_n t/\hbar)+\int_0^{\infty} \mathrm{d} E \psi_E u_E(x) \exp(-\mathrm{i} E t/\hbar),$$
where
##\psi_n=\int_{\mathbb{R}} \mathrm{d} x u_n^*(x) \psi_0(x), \quad \psi_E=\int_{\mathbb{R}} \mathrm{d} x u_E^*(x) \psi_0(x),##
are the matrix elements of the initial wave function wrt. the energy eigenfunctions in the discrete and continuous spectrum of the Hamiltonian.

Gauss quadrature is just a numerical integration technique. You should find tons of such standard routines for integration/quadrature in open-source software libraries.

 

What's unclear with my posting from Dec 28? This really works nicely on a usual personal computer. It's no big deal with a simple numerical-integration routine.

 

Shankar has an excellent section on this.

 

You don't have three solutions but energy eigenstates, defined in the various regions corresponding to your potential. They are all properly matched together by the appropriate boundary conditions and finally make up one function. I don't know, what your course is supposed to teach you, but in any case you should have some idea what the energy eigenfunctions are good for, namely to be able to solve the initial-value problem for the time-dependent Schrödinger equation. After all that's what a big deal of QT is about!