nabla2psi[1:N + 1] = (psi[0:N] + psi[2:N + 2] - 2 * psi[1:N + 1]) / dx2
return 1j * (nabla2psi - V * psi
) # this is the RH of Schroedinger equation!
def d_dt(psi): # find Psi(t+dt)-Psi(t) /dt with 4th order Runge-Kutta method
k1 = f(psi)
k2 = f(psi + dt / 2 * k1)
k3 = f(psi + dt / 2 * k2)
k4 = f(psi + dt * k3)
return (k1 + 2 * k2 + 2 * k3 + k4) / 6
vp = Plotter(interactive=0, axes=2, bg=(0.95, 0.95, 1))
vp.xtitle = ""
vp.ytitle = "|\Psi(x,t)|\^2"
bck = vp.load(datadir + "images/schrod.png").alpha(.3).scale(.0255).pos(
[0, -5, -.1])
barrier = Line(np.stack((x, V * 15, np.zeros_like(x)), axis=1),
c="black",
lw=2)
lines = []
for i in range(0, Nsteps):
for j in range(500):
Psi += d_dt(Psi) * dt # integrate for a while before showing things
A = np.real(Psi * np.conj(Psi)) * 1.5 # psi squared, probability(x)
coords = np.stack((x, A, np.zeros_like(x)), axis=1)
Aline = Line(coords, c="db", lw=3)
vp.show([Aline, barrier, bck])
t = 0
pb = ProgressBar(0, Nsteps, c="blue", ETA=0)
for i in pb.range():
y_eu, v_eu = euler(y_eu, v_eu, t, dt)
y_rk, v_rk = rk4(y_rk, v_rk, t, dt)
t += dt
positions_eu.append(y_eu) # store result of integration
positions_rk.append(y_rk)
pb.print("Integrate: RK-4 and Euler")
####################################################
# Visualize the result
####################################################
settings.useDepthPeeling = False
plt = Plotter(interactive=0, axes=2) # choose axes type nr.2
plt.ytitle = "u(x,t)"
plt.ztitle = "" # will not draw z axis
for i in x:
plt += Point([i, 0, 0], c="green", r=6)
pts_actors_eu = plt.actors # save a copy of the actors list
pts_actors_eu[0].legend = "Euler method"
plt.actors = [] # clean up the list
for i in x:
plt += Point([i, 0, 0], c="red", r=6)
pts_actors_rk = plt.actors # save a copy of the actors list
pts_actors_rk[0].legend = "Runge-Kutta4"
# merge the two lists and set it as the current actors
v_eu, v_rk = np.array(v), np.array(v)
t = 0
pb = ProgressBar(0, Nsteps, c="blue", ETA=0)
for i in pb.range():
y_eu, v_eu = euler(y_eu, v_eu, t, dt)
y_rk, v_rk = rk4(y_rk, v_rk, t, dt)
t += dt
positions_eu.append(y_eu) # store result of integration
positions_rk.append(y_rk)
pb.print("Integrate: RK-4 and Euler")
####################################################
# Visualize the result
####################################################
vp = Plotter(interactive=0, axes=2) # choose axes type nr.2
vp.ytitle = "u(x,t)"
vp.ztitle = "" # will not draw z axis
for i in x:
vp += Point([i, 0, 0], c="green", r=6)
pts_actors_eu = vp.actors # save a copy of the actors list
pts_actors_eu[0].legend = "Euler method"
vp.actors = [] # clean up the list
for i in x:
vp += Point([i, 0, 0], c="red", r=6)
pts_actors_rk = vp.actors # save a copy of the actors list
pts_actors_rk[0].legend = "Runge-Kutta4"
# merge the two lists and set it as the current actors
nabla2psi[1:N + 1] = (psi[0:N] + psi[2:N + 2] - 2 * psi[1:N + 1]) / dx2
return 1j * (nabla2psi - V * psi
) # this is the RH of Schroedinger equation!
def d_dt(psi): # find Psi(t+dt)-Psi(t) /dt with 4th order Runge-Kutta method
k1 = f(psi)
k2 = f(psi + dt / 2 * k1)
k3 = f(psi + dt / 2 * k2)
k4 = f(psi + dt * k3)
return (k1 + 2 * k2 + 2 * k3 + k4) / 6
vp = Plotter(interactive=0, axes=2, bg=(0.95, 0.95, 1))
vp.xtitle = ""
vp.ytitle = "Psi^2(x,t)"
vp.ztitle = ""
bck = vp.load(datadir + "images/schrod.png").alpha(.3).scale(.0255).pos(
[0, -5, -.1])
barrier = Line(np.stack((x, V * 15, np.zeros_like(x)), axis=1),
c="black",
lw=2)
lines = []
for i in range(0, Nsteps):
for j in range(500):
Psi += d_dt(Psi) * dt # integrate for a while before showing things
A = np.real(Psi * np.conj(Psi)) * 1.5 # psi squared, probability(x)
coords = np.stack((x, A, np.zeros_like(x)), axis=1)
Aline = Line(coords, c="db", lw=3)
