# Ap[i,i] = a2
# Ap[i+1,i] = a1 #Bottom
# Ap[i,i+1] = a1 #Right
# Ap[N-2,N-2] = a2
#==============================================================================
# Static plots at time t0
plot(real(psi), label = '$\Psi(x,t)$')
plot(abs(psi), label = '$|\Psi(x,t)|$')
plot(-abs(psi), label = '$-|\Psi(x,t)|$')
xlabel('$x$')
ylabel('$\Psi(x,t)$')
legend()
show()
# Animation
from visual import curve, rate
psi_c = curve()
psi_c.set_x(x-L/2)
#psi = banded(A,v,1,1)
while True:
rate(30)
psi_c.set_y(real(psi)*1e-9)
psi_c.set_z(imag(psi)*1e-9)
for i in range(20):
v = b1*psi[1:N] + b2*(psi[2:N+1] + psi[0:N-1])
psi[1:N] = banded(A,v,1,1)
def homework5_1():
N = int(
input(
"Solving A Chain of Resistors. Please enter the number of unkown junction voltages.\n\n \tN = "
))
A = np.zeros((N, N))
V = 5 #voltage of upper rail
u = 2 #number of non-zero values above diagonal of matrix A
d = u #number of non-zero values below diagonal of matrix A
# populate A in a very ugly way!
for i in range(2, N - 2):
A[i, (i - 2)] = -1
A[i, (i - 1)] = -1
A[i, (i)] = 4
A[i, (i + 1)] = -1
A[i, (i + 2)] = -1
A[0, 0] = 3
A[0, 1] = -1
A[0, 2] = -1
A[1, 0] = -1
A[1, 1] = 4
A[1, 2] = -1
A[1, 3] = -1
A[N - 1, N - 1] = 3
A[N - 1, N - 2] = -1
A[N - 1, N - 3] = -1
A[N - 2, N - 1] = -1
A[N - 2, N - 2] = 4
A[N - 2, N - 3] = -1
A[N - 2, N - 4] = -1
print(A)
w = np.zeros(N)
w[0] = V
w[1] = V
if (N <= 20):
#solve Av=w
v = np.linalg.solve(A, w)
print(v)
if (N > 20):
#reorder matrix A into matrix diagonal ordered form
Anew = np.zeros((1 + u + d, N))
k = 0
for i in range(N):
for j in range(u + 1):
Anew[(u + i - k - j), k] = A[i, k - j]
for l in range(1, d + 1):
if (i < N - l):
Anew[(u + i - k + l), k] = A[i + l, k]
k += 1
print(Anew)
# solve Av=w
v = banded.banded(Anew, w, u, d)
print(v)
#sanity check of all values being less than 5
for i in range(N):
if (v[i] > 5):
print("greater than 5!!!!!")
check = 1
else:
check = 0
if (check == 0):
print("All less than 5!!!!")
for i in range(N):
if (v[i] < 0):
print("less than 0!!!!!")
check = 1
else:
check = 0
if (check == 0):
print("All greater than 0!!!!")
line = ax.plot(x,psi.real)
ax.set_xlabel('$x [m]$',fontsize = 20)
ax.set_ylabel('$\psi$',fontsize = 20)
'''
# BEGIN COMPUTING PSI AND ANIMATING THE RESULT
psi_saves = [] # Array to hold snapshots
t = 0
while t < Nt * h:
line[0].set_ydata(psi.real)
ax.set_title('Time = {0} s'.format(t))
plt.draw()
v = vvec(psi) # Create v vector
psi = banded(
A, v, 1,
1) # Use Gaussian elimination on banded matrices to update psi
if t < eps or abs(t - 1e-16) < eps or abs(t - 1e-15) < eps:
psi_saves.append(psi)
t += h
plt.figure()
plt.subplot(311)
plt.plot(x, psi_saves[0].real)
plt.ylabel('$\psi$', fontsize=20)
plt.title('$0$ s')
plt.subplot(312)
plt.plot(x, psi_saves[1].real)
plt.ylabel('$\psi$', fontsize=20)
plt.title('$10 ^{-16}$ s')
plt.subplot(313)
A = np.zeros((bands, N))
v = np.zeros(N)
v[0] = 5
v[1] = 5
A[0, :] = -1
A[1, :] = -1
A[2, :] = 4
A[2, 0] = 3
A[2, N - 1] = 3
A[3, :] = -1
A[4, :] = -1
V = np.round(b.banded(A, v, 2, 2), decimals=3)
i = np.linspace(1, 6, 6)
print('Voltage at each successive node starting from the first: ', V)
py.scatter(i, V)
py.title('Voltage at each node')
py.show()
#same solution using Gaussian Elimination and backsubsitution to Check
#Example 6.1
import numpy as np
U = 5 A = array([ [3, -1, -1, 0, 0, 0], [-1, 4, -1, -1, 0, 0], [-1,-1, 4, -1, -1, 0], [0, -1,-1, 4, -1, -1], [0, 0, -1,-1, 4, -1], [0, 0, 0, -1, -1, 3] ],float) v = array([U, U, 0, 0, 0, 0],float) x = solve(A,v) print(x) # Banded example N = 10000 Ab = empty((5,N),float) vb = zeros(N,float) vb[[0,1]] = U Ab[:,:] = -1 Ab[2,:] = 4 Ab[2,0] = 3 Ab[2,N-1] = 3 x = banded(Ab,vb,2,2) print(x)
A[0, :] = a2
A[1, :] = a1
A[2:, ] = a2
arr = []
# setup vectors with solely x values
for i in range(len(x)):
arr.append([x[i] - L / 2, 0, 0])
ksi_c = curve(pos=arr)
n = len(arr)
def update_points(ksi_c, y, z):
for i in range(n):
ksi_c.modify(i, y=y[i], z=z[i])
while True:
rate(30)
y = real(ksi) * 1e-9
z = imag(ksi) * 1e-9
# update at each point, keeping x
update_points(ksi_c, y, z)
for i in range(20):
v = b1 * ksi[1:N] + b2 * (ksi[2:N + 1] + ksi[0:N - 1])
ksi[1:N] = banded(A, v, 1, 1)
A = makematrixA(a1,a2,x)
# initialize t
tstart = 0.0 # start time
tend = 1.0e-15 # end time
t = tstart
print_interval = 50
step = 0
print_num = 0
while t < tend:
clf()
# multiply matrix equation v = B * psi
v = rhs(psi,N,b1,b2)
# solve linear system A psi = v for psi
psi=banded(A,v,1,1)
# update animation with psi at current timestep
plot(x, abs(psi),'-b',x,psi.real,'-r',x,-abs(psi),'-b')
plot(x,V(x)/eV/80.0,'g')
ylim((-1,1))
my_title="Time: %e" % (t)
title(my_title)
if step%print_interval==0:
savefig('Lab09-Q2b-%i.png'%print_num)
print_num += 1
step+=1
pause(0.01)
t +=h
#Autor: Pablo Gullith
#Bibliotecas
import numpy as np
import banded as bd
def matriz(N):
A = np.zeros([5, N], float)
for j in range(N - 2):
A[0, j + 2] = A[4, j] = -1
for j in range(N - 1):
A[1, j + 1] = A[3, j] = -1
for j in range(N - 2):
A[2, j + 1] = 4
A[2, 0] = A[2, N - 1] = 3
v = np.zeros(N, float)
v[0] = v[1] = 5
return A, v
A, v = matriz(6)
print("QUANDO N = 6:\n", bd.banded(A, v, 2, 2))
A, v = matriz(10000)
print("QUANDO N = 10000:\n", bd.banded(A, v, 2, 2))
exp(complex(-(x * a - x0)**2 / (2 * sigma**2), kappa * x * a))
for x in range(N + 1)
])
psi[0], psi[N] = 0, 0
# Main program to calculate psi at different time
for i in range(N):
v_tmp = zeros(N + 1, complex)
v_tmp[0] = b1 * psi[0, i] + b2 * (psi[1, i])
for j in range(1, N):
v_tmp[j] = b1 * psi[j, i] + b2 * (psi[j + 1, i] + psi[j - 1, i])
v_tmp[N] = b1 * psi[0, i] + b2 * (psi[N, i])
psi[:, i + 1] = banded.banded(A, v_tmp, 1, 1)
# (b)
# Animation using the method from Lab 08
for i in range(N + 1):
clf()
title("wavelength vs length")
plot(arange(0, N + 1) * a, real(psi[:, i]))
xlabel("length")
ylabel("wavelength")
ylim(-1, 1)
show()
pause(0.01)
# Screen shots at t = t1, t2, t3
# Test with 6 elements
N = 6
U = 5
A = array(
[[3, -1, -1, 0, 0, 0], [-1, 4, -1, -1, 0, 0], [-1, -1, 4, -1, -1, 0],
[0, -1, -1, 4, -1, -1], [0, 0, -1, -1, 4, -1], [0, 0, 0, -1, -1, 3]],
float)
v = array([U, U, 0, 0, 0, 0], float)
x = solve(A, v)
print(x)
# Banded example
N = 10000
Ab = empty((5, N), float)
vb = zeros(N, float)
vb[[0, 1]] = U
Ab[:, :] = -1
Ab[2, :] = 4
Ab[2, 0] = 3
Ab[2, N - 1] = 3
x = banded(Ab, vb, 2, 2)
print(x)
ax = plt.axes()
line = ax.plot(x,psi.real)
ax.set_xlabel('$x [m]$',fontsize = 20)
ax.set_ylabel('$\psi$',fontsize = 20)
'''
# BEGIN COMPUTING PSI AND ANIMATING THE RESULT
psi_saves = [] # Array to hold snapshots
t = 0
while t < Nt*h:
line[0].set_ydata(psi.real)
ax.set_title('Time = {0} s'.format(t))
plt.draw()
v = vvec(psi) # Create v vector
psi = banded(A,v,1,1) # Use Gaussian elimination on banded matrices to update psi
if t < eps or abs(t-1e-16) < eps or abs(t-1e-15) < eps:
psi_saves.append(psi)
t+=h
plt.figure()
plt.subplot(311)
plt.plot(x,psi_saves[0].real)
plt.ylabel('$\psi$',fontsize = 20)
plt.title('$0$ s')
plt.subplot(312)
plt.plot(x,psi_saves[1].real)
plt.ylabel('$\psi$',fontsize = 20)
plt.title('$10 ^{-16}$ s')
plt.subplot(313)
plt.plot(x,psi_saves[2].real)
N = 6 Vp = 5.0 # setup the arrays A = empty([5,N], float) A[0,:] = -1.0 A[1,:] = -1 A[2,:] = 4.0 A[3,:] = -1 A[4,:] = -1.0 A[2,0] = A[2, N-1] = 3.0 v = zeros(N, float) v[0] = v[1] = Vp # solve the equations w = banded(A, v, 2, 2) # make a plot print(w) plot(range(1, N+1), w) plot(range(1, N+1), w, 'ko') ylim(0,5)
A = A(len(xvalues),a1,a2)
#entries for calling banded function
up = 1
down = 1
#start time
t = 0
#end time
tend = 1e-15 #seconds
#an empty list to include si values
si_solutions = []
while t<tend+h:
#solving the matrix equation using banded function
si_new = banded(A,v(si_vals),up,down)
#appending si values to a list
si_solutions.append(si_new)
#making the ew si to be the si used to calculate the newer si
si_vals = si_new
line[0].set_ydata(si_new)
ax.set_title("time={0}".format(t))
plt.ylabel("$\psi$")
plt.draw()
print t
t += h
N = 6 #set up banded matrix A = zeros([5,N],float) A[0,:] = -1 A[1,:] = -1 A[2,:] = 4 A[3,:] = -1 A[4,:] = -1 A[0,0] = 3 A[-1,-1] = 3 w = zeros(N,float) w[0] = V w[1] = V v = banded(A,w,2,2) print v #c) V = 5 #voltage N= 1000 #set up banded matrix A = zeros([5,N],float) A[0,:] = -1 A[1,:] = -1 A[2,:] = 4 A[3,:] = -1 A[4,:] = -1 A[0,0] = 3 A[-1,-1] = 3
A[0,i] = -k
A[1,i] = alpha
A[2,i] = -k
A[1,0] = alpha - k
A[1,N-1] = alpha - k
v = zeros(N,float)
v[0] = C
x = banded(A,v,1,1)
s = []
delta = 2
for i in range(len(x)):
p = (i - len(x)/2)*delta
s.append(vp.sphere(pos = vp.vector(p, 0, 0), radius = 0.3))
t0 = tm.time()
while True:
t = tm.time()
for i in range(len(s)):
p = (i - len(s)/2)*delta + x[i]*sin(omega*(t - t0))
s[i].pos = vp.vector(p, 0, 0)
tm.sleep(0.05)
v = b1 * psi[1:N] + b2 * (psi[2:N + 1] + psi[0:N - 1])
#now to solve the system Ax = v
#x will be time evolved value of psi, A is as given in Newman pg.440
from banded import banded
a1 = 1 + h * 1j * hbar / (2 * M * a**2)
a2 = -h * 1j * hbar / (4 * M * a**2)
A = np.empty([3, N], dtype=complex)
A[0, :] = a2
A[1, :] = a1
A[2, :] = a2
psi_t = np.zeros(N + 1, dtype=complex) #psi at later time t
#Calculate 1 time step:
psi[1:N] = banded(
A, v, 1,
1) #solve for x using Gauss elim and backsub for tridiagnonal matrix
#the above psi_t is the wavefunction on the domain x in [0,L] for t=h (h time step)
#Extend program to perform repeated steps and make an animation
steps = 2000
from pylab import figure, plot, xlabel, ylabel, title, clf, pause, draw
figure()
plot(x, psi.real)
xlabel('$x$ m')
ylabel('$\psi(x,t)$')
title('Wavefunction at time t = 0')
#instantiate emtpy vector for psi(x,t)
from banded import banded
from pylab import plot,show
# Constants
N = 26
C = 1.0
m = 1.0
k = 6.0
omega = 2.0
alpha = 2*k-m*omega*omega
# Set up the initial values of the arrays
A = empty([3,N],float)
for i in range(N):
A[0,i] = -k
A[1,i] = alpha
A[2,i] = -k
A[1,0] = alpha - k
A[1,N-1] = alpha - k
v = zeros(N,float)
v[0] = C
# Solve the equations
x = banded(A,v,1,1)
# Make a plot using both dots and lines
plot(x)
plot(x,"ko")
show()
