Initial commit

.gitignore

@@ -0,0 +1,2 @@
+/,venv
+/.idea

packet.py

@@ -0,0 +1,57 @@
+import numpy as np
+import pylab as pl
+
+x_min = -0.5
+x_max = 0.5
+L = x_max - x_min
+N = 69
+alpha = 0.2
+
+x_vals = np.linspace(x_min, x_max, 10000)
+
+ms = np.arange(-N // 2, N // 2, 1)
+ks = N + ms
+amplitudes = np.exp(-alpha * ms ** 2)
+amplitudes /= np.sum(amplitudes)
+y_vals = np.sum(amplitudes[:, None] * np.cos(ks[:, None] * (2 * np.pi * x_vals / L)), axis=0)
+y_vals /= np.sum(y_vals)
+
+expect_m_sqrd = np.dot(ks ** 2, amplitudes)
+expect_m = np.dot(ks, amplitudes)
+sigma_m = np.sqrt(expect_m_sqrd - expect_m ** 2)
+print(f'Sigma M: {sigma_m}')
+
+diff = np.diff(y_vals)
+minima_indices = np.where(np.r_[True, diff < 0] & np.r_[diff > 0, True])[0]
+maxima_indices = np.where(np.r_[True, diff > 0] & np.r_[diff < 0, True])[0]
+extremes = np.concatenate((minima_indices, maxima_indices))
+envelope_ys = np.abs(y_vals[extremes]) / np.sum(np.abs(y_vals[extremes]))
+envelope_xs = x_vals[extremes]
+expect_x_sqrd = np.dot(envelope_xs ** 2, envelope_ys)
+expect_x = np.dot(envelope_xs, envelope_ys)
+sigma_x = np.sqrt(expect_x_sqrd - expect_x ** 2)
+print(f'Sigma X: {sigma_x}')
+
+print(f'Sigma K * Sigma X = {sigma_m * sigma_x * 2 * np.pi}')
+
+pl.plot(x_vals, np.array(y_vals), label="wave packet")
+pl.plot(x_vals[minima_indices], y_vals[minima_indices], '.', label="minima")
+pl.plot(x_vals[maxima_indices], y_vals[maxima_indices], '.', label="maxima")
+pl.vlines(x=-sigma_x, ymin=np.min(y_vals), ymax=np.max(y_vals), color="red", label="std min")
+pl.vlines(x=sigma_x, ymin=np.min(y_vals), ymax=np.max(y_vals), color="green", label="std max")
+pl.legend()
+pl.title("Wave Packet")
+pl.xlabel("x")
+pl.ylabel("psi")
+pl.grid()
+pl.show()
+
+pl.plot(ms, amplitudes, label="amplitudes")
+pl.vlines(x=-sigma_m, ymin=np.min(amplitudes), ymax=np.max(amplitudes), color="red", label="std min")
+pl.vlines(x=sigma_m, ymin=np.min(amplitudes), ymax=np.max(amplitudes), color="green", label="std max")
+pl.legend()
+pl.title("Fourier Amplitudes")
+pl.xlabel("m")
+pl.ylabel("amplitude")
+pl.grid()
+pl.show()

scratch.py

@@ -0,0 +1,3 @@
+
+# sigma_x = np.sum((y_vals[extremes] * x_vals[extremes]) / np.sum(y_vals[extremes]))
+