h, x, V, R, I = initialize(a, b, N) # initialization
scene = vp.display(background=(1,1,1), ambient=1) # set scene
bars = vpm.bars(x, z, z, z, h, 0.05, (1,0,1)) # wave function
line = vpm.line(x, z, z, (1,0,1), 0.02)
pot = vpm.line(x, V*0.05, z, (0,0,0), 0.02) # potential line
t, ic, cycle = 0.0, 0, 10 # t, animation cycle
dt, psi = 0.001, R + 1j*I # initialize dt, complex psi
ta, pba = [], [] # time, prob. arrays
A = np.ones((3, N+1), dtype=complex) # prepare band matrix A, B
A[1,:] = 2*(2j*h*h/dt - 1 - h*h*V)
dB = - np.conjugate(A[1,:]) # diagonal of B
while(t<=3):
C = dB*psi # prepare RHS
C[1:-1] -= (psi[:-2] + psi[2:])
psi = solve_banded((1,1), A, C) # band matrix solver
t, ic = t+dt, ic+1
if (ic % cycle == 0):
pb = psi.real**2 + psi.imag**2 # probability
ta.append(t), pba.append(pb) # store data
line.move(x, mag*pb, z)
bars.move(x, z, z, mag*pb)
vp.rate(1200), vpm.wait(scene)
(X, Y), ta = np.meshgrid(x, ta), np.array(ta)
plt.figure()
plt.contour(Y, X, pba, 36, linewidths=1)
plt.plot(ta, 5 - ta*ta, 'r--') # classical free fall
plt.xlabel('t (a.u.)'), plt.ylabel('x (a.u.)')
plt.show()
rright = r[:, 0:-1] - r[:, 1:] # rel pos to right neighbor
ftop, fright = force(rtop), force(rright) # forces from top, right
f[0:-1, :] = ftop # force from top
f[:, 0:-1] += fright # force from right
f[1:, :] -= ftop # below, left: use 3rd law
f[:, 1:] -= fright
a = (f - damp*v)/mass + gvec
v[0,0], v[0,-1], v[-1,0], v[-1,-1]=0, 0, 0, 0 # fixed coners
return np.array([v,a])
L, M, N = 2.0, 15, 15 # size, (M,N) particle array
h, mass, damp = 0.01, 0.004, 0.01 # keep damp between [.01,.1]
x, y = np.linspace(0,L,M), np.linspace(0,L,N) # particle grid
r, v = np.zeros((N,M,3)), np.zeros((N,M,3))
spring_k, spring_l = 50.0, x[1]-x[0] # spring const., relaxed length
r[:,:, 0], r[:,:, 1] = np.meshgrid(x,y) # initialize pos
Y, gvec = np.array([r, v]), np.array([0,0,-9.8]) # [v,a], g vector
scene = vp.display(title='Tablecloth', background=(.2,.5,1),
up=(0,0,1), center=(L/2,L/2,-L/4), forward=(1,2,-1))
vp.points(pos=[(0,0,0),(0,L,0),(L,L,0),(L,0,0)], size=50) # corners
x, y, z = r[:,:,0], r[:,:,1], r[:,:,2] # mesh points
net = vpm.net(x, y, z, vp.color.yellow, 0.005) # mesh net
mesh = vpm.mesh(x, y, z, vp.color.red, vp.color.yellow)
while (1):
vp.rate(100), vpm.wait(scene) # pause if key pressed
Y = ode.RK4(cloth, Y, 0, h)
x, y, z = Y[0,:,:,0], Y[0,:,:,1], Y[0,:,:,2]
net.move(x, y, z), mesh.move(x, y, z)
f[1:, :] -= ftop # below, left: use 3rd law
f[:, 1:] -= fright
a = (f - damp * v) / mass + gvec
v[0, 0], v[0, -1], v[-1, 0], v[-1, -1] = 0, 0, 0, 0 # fixed coners
return np.array([v, a])
L, M, N = 2.0, 15, 15 # size, (M,N) particle array
h, mass, damp = 0.01, 0.004, 0.01 # keep damp between [.01,.1]
x, y = np.linspace(0, L, M), np.linspace(0, L, N) # particle grid
r, v = np.zeros((N, M, 3)), np.zeros((N, M, 3))
spring_k, spring_l = 50.0, x[1] - x[0] # spring const., relaxed length
r[:, :, 0], r[:, :, 1] = np.meshgrid(x, y) # initialize pos
Y, gvec = np.array([r, v]), np.array([0, 0, -9.8]) # [v,a], g vector
scene = vp.display(title='Tablecloth',
background=(.2, .5, 1),
up=(0, 0, 1),
center=(L / 2, L / 2, -L / 4),
forward=(1, 2, -1))
vp.points(pos=[(0, 0, 0), (0, L, 0), (L, L, 0), (L, 0, 0)], size=50) # corners
x, y, z = r[:, :, 0], r[:, :, 1], r[:, :, 2] # mesh points
net = vpm.net(x, y, z, vp.color.yellow, 0.005) # mesh net
mesh = vpm.mesh(x, y, z, vp.color.red, vp.color.yellow)
while (1):
vp.rate(100), vpm.wait(scene) # pause if key pressed
Y = ode.RK4(cloth, Y, 0, h)
x, y, z = Y[0, :, :, 0], Y[0, :, :, 1], Y[0, :, :, 2]
net.move(x, y, z), mesh.move(x, y, z)
rij = r[i]-r[i+1:] # rij for all j>i
rij[rij > HL] -= L # periodic bc
rij[rij < -HL] += L
r2 = np.sum(rij*rij, axis=1) # |rij|^2
r6 = r2*r2*r2
for k in [0,1,2]: # L-J force in x,y,z
fij = 12.*(1. - r6)*rij[:,k]/(r6*r6*r2)
a[i,k] += np.sum(fij)
a[i+1:,k] -= fij # 3rd law
return a
L, N = 10.0, 32 # cube size, num. atoms
atoms, HL, t, h = [], L/2., 0., 0.002
r, v = np.zeros((N,3)), np.zeros((N,3))
scene = vp.display(background=(.2,.5,1), center=(L/2, L/3, L/2))
vp.box(pos=(HL,HL,HL), length=L, height=L, width=L, opacity=0.3)
for i in range(N): # initial pos, vel
for k in range(3):
r[i,k] = L*rnd.random()
v[i,k] = 1-2*rnd.random()
atoms.append(vp.sphere(pos=r[i], radius=0.04*L, color=(1,0,1)))
v -= np.sum(v, axis=0)/N # center of mass frame
while (1):
vpm.wait(scene), vp.rate(1000)
r, v = ode.leapfrog(nbody, r, v, t, h)
r[r > L] -= L # periodic bc
r[r < 0.] += L
for i in range(N): atoms[i].pos = r[i] # move atoms
return [psi[1], 2*(V(x)-E)*psi[0]]
# initialization and animation setup
a, V0 = 4.0, 4. # well width, depth
R, N = 4*a, 200 # limit, intervals
xa = np.linspace(-R, R, 2*N+1) # grid
h, z = xa[1]-xa[0], np.zeros(2*N+1) # step size
E, dE, dpsi, psix = -V0, 0.001, 1.0, np.zeros(2*N+1)
scene = vp.display(background=(.2,.5,1), range=1.5*a)
wf = vpm.line(xa, psix, z, vp.color.red, .05)
pot = vpm.line(xa, .5*np.vectorize(V)(xa), z, (1,1,1), .04) # pot. V
info = vp.label(pos=(0, -0.6*a, 0), box=False, height=20)
while (E < 0.0):
psi, x = [.0, .1], -R
for i in range(N): # WF for x <=0
psi = ode.RK45n(sch, psi, x, h)
x += h
psix[i+1] = psi[0]
psix[N+1:] = psix[N-1::-1] # WF for x > 0 by reflection
if (dpsi*psi[1] < 0.): # dpsi/dx changes sign
info.text='Energy found, E=%5.4f' %(E-dE/2)
vpm.pause(scene) # any key to continue
else:
info.text='E=%5.3f' %(E)
wf.move(xa, 2*psix/max(psix), z), vpm.wait(scene), vp.rate(2000)
dpsi = psi[1] # old dpsi/dx at E
E += dE
vp.arrow(pos=(.2, -.2, L), axis=ay, length=0.2, color=(0,1,1))
vp.arrow(pos=(.2, -.2, L), axis=az, length=0.2, color=(1,1,1))
vp.label(pos=(.45, -.2, L), text='E', box=False, height=30)
vp.label(pos=(.25, -.0, L), text='B', box=False, height=30)
vp.label(pos=(.2, -.15, L+.3), text='v', box=False, height=30)
idx, z = np.arange(n), np.linspace(-L, 2*L, n) # order of vectors
mag = scale*np.sin(2*np.pi*z/L) # sine envelope
ewave = vp.curve(color=(1,1,0), pos=np.column_stack((mag,0*z,z)))
bwave = vp.curve(color=(0,1,1), pos=np.column_stack((0*z,mag,z)))
for i in idx:
E.append( vp.arrow(pos=(0, 0, z[i]), axis=ax, length=mag[i],
color=(1,1,0)) )
B.append( vp.arrow(pos=(0, 0, z[i]), axis=ay, length=mag[i],
color=(0,1,1)) )
while True:
vp.rate(100), vpm.wait(scene) # hit a key to pause
t, mg = t + dt, mag*np.cos(t) # sinusoidal wave
for i in range(n):
E[i].pos.z += v*dt # traveling wave
B[i].pos.z += v*dt
if (E[i].pos.z > 2*L): # wrap around
E[i].pos.z, B[i].pos.z = -L, -L
idx = np.insert(idx, 0, i)[:-1] # move to end
E[i].axis, B[i].axis = ax, ay # reset axis to
E[i].length, B[i].length = mg[i], mg[i] # draw correctly
id = idx[i]
ewave.pos[i] = (mg[id], 0, E[id].pos.z) # envelope curves
bwave.pos[i] = (0, mg[id], B[id].pos.z)
h, x, V, R, I = initialize(a, b, N) # initialization
scene = vp.display(background=(1, 1, 1), ambient=1) # set scene
bars = vpm.bars(x, z, z, z, h, 0.05, (1, 0, 1)) # wave function
line = vpm.line(x, z, z, (1, 0, 1), 0.02)
pot = vpm.line(x, V * 0.05, z, (0, 0, 0), 0.02) # potential line
t, ic, cycle = 0.0, 0, 10 # t, animation cycle
dt, psi = 0.001, R + 1j * I # initialize dt, complex psi
ta, pba = [], [] # time, prob. arrays
A = np.ones((3, N + 1), dtype=complex) # prepare band matrix A, B
A[1, :] = 2 * (2j * h * h / dt - 1 - h * h * V)
dB = -np.conjugate(A[1, :]) # diagonal of B
while (t <= 3):
C = dB * psi # prepare RHS
C[1:-1] -= (psi[:-2] + psi[2:])
psi = solve_banded((1, 1), A, C) # band matrix solver
t, ic = t + dt, ic + 1
if (ic % cycle == 0):
pb = psi.real**2 + psi.imag**2 # probability
ta.append(t), pba.append(pb) # store data
line.move(x, mag * pb, z)
bars.move(x, z, z, mag * pb)
vp.rate(1200), vpm.wait(scene)
(X, Y), ta = np.meshgrid(x, ta), np.array(ta)
plt.figure()
plt.contour(Y, X, pba, 36, linewidths=1)
plt.plot(ta, 5 - ta * ta, 'r--') # classical free fall
plt.xlabel('t (a.u.)'), plt.ylabel('x (a.u.)')
plt.show()
vp.arrow(pos=(.2, -.2, L), axis=ay, length=0.2, color=(0, 1, 1))
vp.arrow(pos=(.2, -.2, L), axis=az, length=0.2, color=(1, 1, 1))
vp.label(pos=(.45, -.2, L), text='E', box=False, height=30)
vp.label(pos=(.25, -.0, L), text='B', box=False, height=30)
vp.label(pos=(.2, -.15, L + .3), text='v', box=False, height=30)
idx, z = np.arange(n), np.linspace(-L, 2 * L, n) # order of vectors
mag = scale * np.sin(2 * np.pi * z / L) # sine envelope
ewave = vp.curve(color=(1, 1, 0), pos=np.column_stack((mag, 0 * z, z)))
bwave = vp.curve(color=(0, 1, 1), pos=np.column_stack((0 * z, mag, z)))
for i in idx:
E.append(
vp.arrow(pos=(0, 0, z[i]), axis=ax, length=mag[i], color=(1, 1, 0)))
B.append(
vp.arrow(pos=(0, 0, z[i]), axis=ay, length=mag[i], color=(0, 1, 1)))
while True:
vp.rate(100), vpm.wait(scene) # hit a key to pause
t, mg = t + dt, mag * np.cos(t) # sinusoidal wave
for i in range(n):
E[i].pos.z += v * dt # traveling wave
B[i].pos.z += v * dt
if (E[i].pos.z > 2 * L): # wrap around
E[i].pos.z, B[i].pos.z = -L, -L
idx = np.insert(idx, 0, i)[:-1] # move to end
E[i].axis, B[i].axis = ax, ay # reset axis to
E[i].length, B[i].length = mg[i], mg[i] # draw correctly
id = idx[i]
ewave.pos[i] = (mg[id], 0, E[id].pos.z) # envelope curves
bwave.pos[i] = (0, mg[id], B[id].pos.z)
rij[rij < -HL] += L
r2 = np.sum(rij * rij, axis=1) # |rij|^2
r6 = r2 * r2 * r2
for k in [0, 1, 2]: # L-J force in x,y,z
fij = 12. * (1. - r6) * rij[:, k] / (r6 * r6 * r2)
a[i, k] += np.sum(fij)
a[i + 1:, k] -= fij # 3rd law
return a
L, N = 10.0, 32 # cube size, num. atoms
atoms, HL, t, h = [], L / 2., 0., 0.002
r, v = np.zeros((N, 3)), np.zeros((N, 3))
scene = vp.display(background=(.2, .5, 1), center=(L / 2, L / 3, L / 2))
vp.box(pos=(HL, HL, HL), length=L, height=L, width=L, opacity=0.3)
for i in range(N): # initial pos, vel
for k in range(3):
r[i, k] = L * rnd.random()
v[i, k] = 1 - 2 * rnd.random()
atoms.append(vp.sphere(pos=r[i], radius=0.04 * L, color=(1, 0, 1)))
v -= np.sum(v, axis=0) / N # center of mass frame
while (1):
vpm.wait(scene), vp.rate(1000)
r, v = ode.leapfrog(nbody, r, v, t, h)
r[r > L] -= L # periodic bc
r[r < 0.] += L
for i in range(N):
atoms[i].pos = r[i] # move atoms
return [psi[1], 2 * (V(x) - E) * psi[0]]
# initialization and animation setup
a, V0 = 4.0, 4. # well width, depth
R, N = 4 * a, 200 # limit, intervals
xa = np.linspace(-R, R, 2 * N + 1) # grid
h, z = xa[1] - xa[0], np.zeros(2 * N + 1) # step size
E, dE, dpsi, psix = -V0, 0.001, 1.0, np.zeros(2 * N + 1)
scene = vp.display(background=(.2, .5, 1), range=1.5 * a)
wf = vpm.line(xa, psix, z, vp.color.red, .05)
pot = vpm.line(xa, .5 * np.vectorize(V)(xa), z, (1, 1, 1), .04) # pot. V
info = vp.label(pos=(0, -0.6 * a, 0), box=False, height=20)
while (E < 0.0):
psi, x = [.0, .1], -R
for i in range(N): # WF for x <=0
psi = ode.RK45n(sch, psi, x, h)
x += h
psix[i + 1] = psi[0]
psix[N + 1:] = psix[N - 1::-1] # WF for x > 0 by reflection
if (dpsi * psi[1] < 0.): # dpsi/dx changes sign
info.text = 'Energy found, E=%5.4f' % (E - dE / 2)
vpm.pause(scene) # any key to continue
else:
info.text = 'E=%5.3f' % (E)
wf.move(xa, 2 * psix / max(psix), z), vpm.wait(scene), vp.rate(2000)
dpsi = psi[1] # old dpsi/dx at E
E += dE
amnt = amn0*np.outer(phase, phase) # $a_{mn}(t)=a_{mn}(0)e^{-i E_{mn} t}$
for m in range(N):
s = 0.0
for n in range(N): s += amnt[m,n]*uny[n] # vector operation
wf += s*umx[m] # $\sum a_{mn} u_m(x) u_n(y)$
return wf
t, num, plot = 0., 1, False # set plot=False to animate forever
time = [0.0, 0.5, 0.9, 1.4, 2.0, 3.0, 40, 63.7, 64.2] # snapshots
a, N, En, amn0, X, Y, umx, uny = initialize()
scene = vp.display(background=(.2,.5,1), center=(a/2,a/2,-0.5),
up=(0,0,1), forward=(1,2,-1))
mesh = vpm.mesh(X, Y, 0*X, vp.color.yellow, vp.color.red)
info = vp.label(pos=(a/4, .8*a,1), height=20)
while True:
wf = psi(amn0, t)
mesh.move(X, Y, wf.real*3) # show real part, times 3
info.text='%5.2f' %(t)
vpm.wait(scene), vp.rate(40)
if (plot):
plt.subplot(3, 3, num)
plt.imshow(np.abs(wf)**2, cmap=plt.cm.jet) # try contourf()
plt.xticks([],[]), plt.yticks([],[]) # omit ticks
plt.xlabel('t=%s' %(t))
if (num > len(time)-1): break
t, num = time[num], num+1
else:
t = t + 0.05
if (plot): plt.show()
wf += s * umx[m] # $\sum a_{mn} u_m(x) u_n(y)$
return wf
t, num, plot = 0., 1, False # set plot=False to animate forever
time = [0.0, 0.5, 0.9, 1.4, 2.0, 3.0, 40, 63.7, 64.2] # snapshots
a, N, En, amn0, X, Y, umx, uny = initialize()
scene = vp.display(background=(.2, .5, 1),
center=(a / 2, a / 2, -0.5),
up=(0, 0, 1),
forward=(1, 2, -1))
mesh = vpm.mesh(X, Y, 0 * X, vp.color.yellow, vp.color.red)
info = vp.label(pos=(a / 4, .8 * a, 1), height=20)
while True:
wf = psi(amn0, t)
mesh.move(X, Y, wf.real * 3) # show real part, times 3
info.text = '%5.2f' % (t)
vpm.wait(scene), vp.rate(40)
if (plot):
plt.subplot(3, 3, num)
plt.imshow(np.abs(wf)**2, cmap=plt.cm.jet) # try contourf()
plt.xticks([], []), plt.yticks([], []) # omit ticks
plt.xlabel('t=%s' % (t))
if (num > len(time) - 1): break
t, num = time[num], num + 1
else:
t = t + 0.05
if (plot):
plt.show()
c = 1.0/np.sqrt(s*np.sqrt(np.pi))
return c*np.exp(-((x-x0)/s)**2/2)
def initialize(a, b, N): # set parameters
x = np.linspace(a, b, N+1) # grid points
V = 0.5*x*x # SHO potential
s, x0 = 0.5, -5.0 # width (sigma), center of gaussian
R, I = gaussian(s, x0, x), np.zeros(N+1) # real, imag w.f.
return x[1]-x[0], x,V,R,I # grid size, h=x[1]-x[0]
a, b, N = -10., 10., 500 # space range [a,b], num. intervals
z, mag = np.zeros(N+1), 4 # zeros, magnifying factor
h, x, V, R, I = initialize(a, b, N) # initialization
scene = vp.display(background=(1,1,1), ambient=1) # set scene
bars = vpm.bars(x, z, z, z, h, 0.05, (1,0,1)) # wave function
line = vpm.line(x, z, z, (1,0,1), 0.02)
pot = vpm.line(x, V*0.1, z, (0,0,0), 0.02) # potential line
t, ic, cycle = 0.0, 0, 20 # t, animation cycle
dt = h*h*0.5 # time step, dt < h*h
while True:
R, I = ode.leapfrog(sch_eqn, R, I, t, dt) # main work
ic += 1
if (ic % cycle == 0): # animate
pb = R*R + I*I # probability
line.move(x, mag*pb, z)
bars.move(x, z, z, mag*pb)
vp.rate(3000), vpm.wait(scene)
return c * np.exp(-((x - x0) / s)**2 / 2)
def initialize(a, b, N): # set parameters
x = np.linspace(a, b, N + 1) # grid points
V = 0.5 * x * x # SHO potential
s, x0 = 0.5, -5.0 # width (sigma), center of gaussian
R, I = gaussian(s, x0, x), np.zeros(N + 1) # real, imag w.f.
return x[1] - x[0], x, V, R, I # grid size, h=x[1]-x[0]
a, b, N = -10., 10., 500 # space range [a,b], num. intervals
z, mag = np.zeros(N + 1), 4 # zeros, magnifying factor
h, x, V, R, I = initialize(a, b, N) # initialization
scene = vp.display(background=(1, 1, 1), ambient=1) # set scene
bars = vpm.bars(x, z, z, z, h, 0.05, (1, 0, 1)) # wave function
line = vpm.line(x, z, z, (1, 0, 1), 0.02)
pot = vpm.line(x, V * 0.1, z, (0, 0, 0), 0.02) # potential line
t, ic, cycle = 0.0, 0, 20 # t, animation cycle
dt = h * h * 0.5 # time step, dt < h*h
while True:
R, I = ode.leapfrog(sch_eqn, R, I, t, dt) # main work
ic += 1
if (ic % cycle == 0): # animate
pb = R * R + I * I # probability
line.move(x, mag * pb, z)
bars.move(x, z, z, mag * pb)
vp.rate(3000), vpm.wait(scene)