Quantum tunneling through a sharp and soft step

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Not very exciting and novel, but my first upload here 7 years ago called for a smoother 4k version. I will upload a MWE Python code soon for those interested.

Music: Reuse of a song from another of my videos. By @gpcbass of course. The beautiful piece is called Vanessa.

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Python code:

import matplotlib.pyplot as plt
from matplotlib.patches import Rectangle
import numpy as N
import scipy.linalg as SLA
import matplotlib as mpl
import os, gc

= 16/9*6, 6

ymin = -2
xmin = ymin*16/9
xmax = abs(xmin);ymax = abs(ymin)

path = 'DestDir' #Frame directory

NN = 1000 #number of x coordinates
pwp = 40 #momentum
sig = .15 #width of wave packet
Vmax= pwp**2/2*1. #height of tunneling barrier
frames = 200 #number of frames

dx = (2*xmax-2*xmin)/NN
xvec = N.arange(xmin*2, xmax*2, dx)
b = -1/2/dx**2

#Finite difference Laplacian
diag = -N.diagflat(2*b*N.ones(NN))
offdiag = N.diagflat(b*N.ones(NN-1), 1)
D2 =
pot = N.zeros(NN)
for ii in range(NN):
if (ii>int(.495*NN) and ii<int(.505*NN)):
pot[ii] = Vmax
H = D2+N.diagflat(pot)

ev, evec = SLA.eigh(H)
idx = ev.argsort()
ev, evec = ev[idx], evec[idx]

initwf = #initial Gaussian
init = ev[0]
cm = N.array([N.sum(initwf*evec[:, mm]) for mm in range(len(ev))])
cm = cm/SLA.norm(cm)
psi = [cm[jj]*evec[:, jj] for jj in range(len(ev))]

fig = plt.figure()
for tt in range(frames):
tt2 = tt/frames
Psi = psi[0]*N.exp(1j*ev[0]*tt2)
for jj in range(1, len(psi)):

plt.xlim([xmin, xmax])
plt.ylim([ymin*.3, ymax*.3])
plt.gca().add_patch(Rectangle((-4, -10), 10, 10, linewidth=1, edgecolor=None, facecolor='gray'))
if max(pot)>0:
plt.fill(xvec, pot/abs(b)*.5, linewidth=1, color='gray')
else:
plt.fill(xvec, pot/abs(b)*.5, linewidth=1, color='black')

plt.plot(xvec, 5*N.abs(Psi)**2, linewidth=1, color='red')
plt.savefig(path+'/fig'+f"{tt:04d}.png",
bbox_inches='tight', pad_inches=0)
plt.clf()
plt.close("fig")

animations_ag