Numerov animation

numerov-2-ani.py

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+import pylab as pl
+import numpy as np
+import matplotlib.animation as animation
+
+iterations = 60000  # iterations for approximation
+
+step = 0.0001  # step size for x
+step_sqrd = pow(step, 2)  # square of step size
+
+n = 2
+epsilon = (n * np.pi) ** 2 / (12 ** 2)  # energy level, should be integer n+1/2 for good solutions
+
+psi = 0.0  # initial value of wave function
+potential = 0.0  # initial value of the potential energy function
+pos = -1 * (iterations - 2) * step  # initial value of the position
+
+potential_past_2 = 0  # epsilon + pos - 2 * step  # k_0, potential energy at two steps before current
+potential_past_1 = 0  # epsilon + pos - step  # k_1, potential energy at one step before current
+amplitude = 0.1  # initial amplitude of wave function
+psi_past_2 = 0  # y_0, wave function at two steps before current
+psi_past_1 = amplitude  # amplitude  # y_1, wave function at one step before current
+
+x_out = []  # list to store x values for plotting
+y_out = []  # list to store y values for plotting
+
+count = -1 * iterations + 2  # counter for the loop
+
+fig, ax = pl.subplots()
+ax.set_xlim(-6, 6)
+ax.set_ylim(-1, 1)
+line, = ax.plot(x_out, y_out, label=f'epsilon = {epsilon}')
+
+
+def update(frame):
+    global pos, potential_past_1, psi_past_1, potential_past_2, psi_past_2, ax
+    # Numerov integration loop
+    # count += 1
+    for i in range(0, 1000, 1):
+        pos += step
+        potential = epsilon
+        # potential = epsilon  # potential energy function
+        b = step_sqrd / 12  # constant used for Numerov
+        # Numerov method to calculate psi at current step
+        psi = ((2 * (1 - 5 * b * potential_past_1) * psi_past_1 - (1 + b * potential_past_2) * psi_past_2)
+               / (1 + b * potential))
+
+        # Save for plotting
+        x_out.append(pos)
+        y_out.append(psi)
+
+        # Shift for next iteration
+        psi_past_2 = psi_past_1
+        psi_past_1 = psi
+        potential_past_2 = potential_past_1
+        potential_past_1 = potential
+
+    ax.set_xlim(min(x_out), max(x_out))
+    ax.set_ylim(min(y_out), max(y_out))
+    line.set_data(x_out, y_out)
+    return line,
+
+
+# Plot
+ani = animation.FuncAnimation(fig, update, frames=np.arange((-1 * iterations + 2) // 1000, (iterations + 2) // 1000),
+                              blit=True, repeat=False)
+pl.show()

numerov-ani.py

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+import pylab as pl
+import numpy as np
+import matplotlib.animation as animation
+
+iterations = 60000  # iterations for approximation
+
+step = 0.0001  # step size for x
+step_sqrd = pow(step, 2)  # square of step size
+
+epsilon = 2.5  # energy level, should be integer n+1/2 for good solutions
+
+psi = 0.0  # initial value of wave function
+potential = 0.0  # initial value of the potential energy function
+pos = -1 * (iterations - 2) * step  # initial value of the position
+
+potential_past_2 = epsilon + pos - 2 * step  # k_0, potential energy at two steps before current
+potential_past_1 = epsilon + pos - step  # k_1, potential energy at one step before current
+amplitude = 0.1  # initial amplitude of wave function
+psi_past_2 = 0  # y_0, wave function at two steps before current
+psi_past_1 = amplitude  # y_1, wave function at one step before current
+
+x_out = []  # list to store x values for plotting
+y_out = []  # list to store y values for plotting
+
+count = -1 * iterations + 2  # counter for the loop
+
+fig, ax = pl.subplots()
+ax.set_xlim(-6, 6)
+ax.set_ylim(-1, 1)
+line, = ax.plot(x_out, y_out, label=f'epsilon = {epsilon}')
+
+
+def update(frame):
+    global pos, potential_past_1, psi_past_1, potential_past_2, psi_past_2, ax
+    # Numerov integration loop
+    # count += 1
+    for i in range(0, 1000, 1):
+        pos += step
+        potential = 2 * epsilon - pow(pos, 2)  # potential energy function
+        b = step_sqrd / 12  # constant used for Numerov
+        # Numerov method to calculate psi at current step
+        psi = ((2 * (1 - 5 * b * potential_past_1) * psi_past_1 - (1 + b * potential_past_2) * psi_past_2)
+               / (1 + b * potential))
+
+        # Save for plotting
+        x_out.append(pos)
+        y_out.append(psi)
+
+        # Shift for next iteration
+        psi_past_2 = psi_past_1
+        psi_past_1 = psi
+        potential_past_2 = potential_past_1
+        potential_past_1 = potential
+
+    ax.set_xlim(min(x_out), max(x_out))
+    ax.set_ylim(min(y_out), max(y_out))
+    line.set_data(x_out, y_out)
+    return line,
+
+
+# Plot
+ani = animation.FuncAnimation(fig, update, frames=np.arange((-1 * iterations + 2) // 1000, (iterations - 2) // 1000),
+                              blit=True, repeat=False)
+pl.show()

numerov-animation.py

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+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.animation as animation
+
+iterations = 60000  # iterations for approximation
+step = 0.0001  # step size for x
+step_sqrd = pow(step, 2)  # square of step size
+epsilon = 2.5  # energy level, should be integer n+1/2 for good solutions
+
+psi = 0.0  # initial value of wave function
+potential = 0.0  # initial value of the potential energy function
+pos = -1 * (iterations - 2) * step  # initial value of the position
+
+potential_past_2 = epsilon + pos - 2 * step  # k_0, potential energy at two steps before current
+potential_past_1 = epsilon + pos - step  # k_1, potential energy at one step before current
+amplitude = 0.1  # initial amplitude of wave function
+psi_past_2 = 0  # y_0, wave function at two steps before current
+psi_past_1 = amplitude  # y_1, wave function at one step before current
+
+x_out = []  # list to store x values for plotting
+y_out = []  # list to store y values for plotting
+
+fig, ax = plt.subplots()
+line, = ax.plot([], [], label=f'epsilon = {epsilon}')
+
+ax.set_xlim(-1, 1)
+ax.set_ylim(-1, 1)
+ax.set_xlabel("x")
+ax.set_ylabel("y")
+ax.set_title("Schrodinger Eqn in Harmonic Potential")
+ax.legend(loc=1)
+
+
+def init():
+    line.set_data([], [])
+    return line,
+
+
+def update(frame):
+    global pos, potential, psi, potential_past_2, potential_past_1, psi_past_2, psi_past_1
+    for i in range(frame * 1000, (frame * 1000) + 1000):
+        print(i)
+        pos += step
+        potential = 2 * epsilon - pow(pos, 2)  # potential energy function
+        b = step_sqrd / 12  # constant used for Numerov
+        # Numerov method to calculate psi at current step
+        psi = ((2 * (1 - 5 * b * potential_past_1) * psi_past_1 - (1 + b * potential_past_2) * psi_past_2)
+               / (1 + b * potential))
+
+        # Save for plotting
+        x_out.append(pos)
+        y_out.append(psi)
+
+        # Shift for next iteration
+        psi_past_2 = psi_past_1
+        psi_past_1 = psi
+        potential_past_2 = potential_past_1
+        potential_past_1 = potential
+
+    line.set_data(x_out, y_out)
+    return line,
+
+
+ani = animation.FuncAnimation(fig, update, frames=np.arange(0, (iterations - 2)), init_func=init, blit=True)
+plt.show()

numerov-thing.py

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+import pylab as lab
+import numpy as np
+
+iterations = 60000  # iterations for approximation
+
+step = 0.0001  # step size for x
+step_sqrd = pow(step, 2)  # square of step size
+
+well_width = 2
+
+# try values 1, 2, and 3 for various solutions
+n = 1
+epsilon = (n * np.pi) ** 2 / (well_width ** 2)  # energy level, should be integer n+1/2 for good solutions
+
+psi = 0.0  # initial value of wave function
+potential = 0.0  # initial value of the potential energy function
+pos = -1 * (iterations - 2) * step  # initial value of the position
+
+potential_past_2 = epsilon + pos - 2 * step  # k_0, potential energy at two steps before current
+potential_past_1 = epsilon + pos - step  # k_1, potential energy at one step before current
+amplitude = 0.1  # initial amplitude of wave function
+psi_past_2 = 0  # y_0, wave function at two steps before current
+psi_past_1 = amplitude  # y_1, wave function at one step before current
+
+x_out = []  # list to store x values for plotting
+y_out = []  # list to store y values for plotting
+
+count = -1 * iterations + 2  # counter for the loop
+
+# Numerov integration loop
+while count < iterations - 2:
+    count += 1
+    pos += step
+    # close enough to infinity :)
+    potential = 100000000
+    if abs(pos) < well_width / 2:
+        potential = epsilon  # potential energy function
+    b = step_sqrd / 12  # constant used for Numerov
+    # Numerov method to calculate psi at current step
+    psi = ((2 * (1 - 5 * b * potential_past_1) * psi_past_1 - (1 + b * potential_past_2) * psi_past_2)
+           / (1 + b * potential))
+
+    # Save for plotting
+    x_out.append(pos)
+    y_out.append(psi)
+
+    # Shift for next iteration
+    psi_past_2 = psi_past_1
+    psi_past_1 = psi
+    potential_past_2 = potential_past_1
+    potential_past_1 = potential
+
+# Plot
+lab.figure(1)
+lab.plot(x_out, y_out, label=f'epsilon = {epsilon}')
+lab.xlabel("x")
+lab.ylabel("y")
+lab.title("Schrodinger Eqn in Harmonic Potential")
+lab.legend(loc=1)
+lab.show()