Add ps11-2.14

ps11-2.14.py

@@ -0,0 +1,70 @@
+import numpy as np
+import pylab as pl
+
+
+def ladder_plus(xs: np.ndarray, psi: np.ndarray) -> np.ndarray:
+    dx = xs[1] - xs[0]
+    return -np.gradient(psi, dx) + xs * psi
+
+
+def ladder_minus(xs: np.ndarray, psi: np.ndarray) -> np.ndarray:
+    dx = xs[1] - xs[0]
+    return np.gradient(psi, dx) + xs * psi
+
+
+def normalize(psi: np.ndarray) -> None:
+    psi /= np.sqrt(np.sum(psi * np.conjugate(psi)))
+
+
+x_vals = np.linspace(-5, 5, 1000)
+dx = x_vals[1] - x_vals[0]
+
+psi_0 = np.exp(-x_vals ** 2)
+normalize(psi_0)
+
+psi_funcs = [psi_0]
+for i in range(1, 4):
+    psi_next = ladder_plus(x_vals, psi_funcs[i - 1])
+    normalize(psi_next)
+    psi_funcs.append(psi_next)
+
+print("==========")
+expect_xs = []
+for i, psi in enumerate(psi_funcs):
+    expect_xs.append(np.sum(np.conjugate(psi) * x_vals * psi))
+    print(f'<x>_{i} = {expect_xs[i]}')
+
+print("==========")
+expect_ps = []
+for i, psi in enumerate(psi_funcs):
+    expect_ps.append(np.sum(np.conjugate(psi) * -np.gradient(psi, dx)))
+    print(f'<p>_{i} = {expect_ps[i]}')
+
+print("==========")
+expect_x_sqrds = []
+for i, psi in enumerate(psi_funcs):
+    expect_x_sqrds.append(np.sum(np.conjugate(psi) * x_vals * x_vals * psi))
+    print(f'<x^2>_{i} = {expect_x_sqrds[i]}')
+
+print("==========")
+expect_p_sqrds = []
+for i, psi in enumerate(psi_funcs):
+    expect_p_sqrds.append(np.sum(np.conjugate(psi) * -np.gradient(np.gradient(psi, dx), dx)))
+    print(f'<p^2>_{i} = {expect_p_sqrds[i]}')
+
+print("==========")
+sigma_xs = []
+for i, psi in enumerate(psi_funcs):
+    sigma_xs.append(np.sqrt(expect_x_sqrds[i] - expect_xs[i] ** 2))
+    print(f'sigma_x_{i} = {sigma_xs[i]}')
+
+print("==========")
+sigma_ps = []
+for i, psi in enumerate(psi_funcs):
+    sigma_ps.append(np.sqrt(expect_p_sqrds[i] - expect_ps[i] ** 2))
+    print(f'sigma_p_{i} = {sigma_ps[i]}')
+
+print("==========")
+print("hbar/2 = 0.5")
+for i, psi in enumerate(psi_funcs):
+    print(f'sigma_x_{i} * sigma_p_{i} = {sigma_xs[i] * sigma_ps[i]}')