Add ps12-3

ps12-3.py

@@ -0,0 +1,33 @@
+import numpy as np
+import pylab as pl
+
+
+# recurrence relation is H_n(x) = 2*x*H_{n-1}(x) - 2*(n-1)*H_{n-2}(x)
+# source https://en.wikipedia.org/wiki/Hermite_polynomials
+def hermite(x: np.ndarray, n: int) -> np.ndarray:
+    if n == 0:
+        return 1.0 + 0.0 * x
+    if n == 1:
+        return 2.0 * x
+    return 2.0 * x * hermite(x, n - 1) - 2.0 * (n - 1) * hermite(x, n - 2)
+
+
+def normalize(psi: np.ndarray) -> None:
+    psi /= np.sqrt(np.sum(psi * np.conjugate(psi)))
+
+
+x_vals = np.linspace(-5, 5, 1000)
+gaussian = np.exp(-x_vals ** 2)
+
+psi_funcs = []
+for n in range(4):
+    psi = hermite(x_vals, n) * gaussian
+    normalize(psi)
+    psi_funcs.append(psi)
+
+for i, psi in enumerate(psi_funcs):
+    psi_sqrd = np.abs(psi) ** 2
+    # integrate in the region outside of -sqrt(2):sqrt(2)
+    prob = (np.trapz(psi_sqrd[x_vals < -np.sqrt(2)], x_vals[x_vals < -np.sqrt(2)])
+            + np.trapz(psi_sqrd[x_vals > np.sqrt(2)], x_vals[x_vals > np.sqrt(2)]))
+    print(f'prob_outside_{i} = {prob}')