{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "A particle moves in a periodic box of length $L=10$. Consider a position-space wavefunction $\\psi(x)$ which is unity in the interval $x_1=2\\le x \\le x_2=3$ and zero for all other values of $x$. Approximate this wave function as a Fourier series with $2\\ell+1$ terms with $-\\ell \\le m \\le \\ell$ and plot the result for $\\ell$ =10, 20, 40, 80, 100 and 200." ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "image/png": 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POSUaQThlSuTajRrVfrTnoEH+HKt8xxk4MFo//nhd2npb8W1HHBFty87/6+wYSUJhRUgq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\nwMKhsCIkcTr9BTLGtAC4CsA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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from scipy.integrate import quad\n", "# assign constants\n", "L=10 # size of region in which particle moves.\n", "x2=3 # upper limit of psi(x)\n", "x1=2 # lower limit of psi(x)\n", "l_max=200 #(-l \\le m \\le +l) largest value of l used below\n", "Dx = 0.01 # Accuracy for plotting position\n", "Nx = int(L/Dx)\n", "psi_tilde = np.zeros((2*l_max+1),dtype=complex)\n", "# Calculate psi_tilde for many values of m, not all of which must be used.\n", "for m in range(-l_max,l_max+1):\n", " fr = lambda x: np.cos(2*np.pi*m*x/L) # Real part of given wave function multiplied by inversse Fourier factor.\n", " fi = lambda x: np.sin(2*np.pi*m*x/L) # Imaginary part of given wave function multiplied by inversse Fourier factor.\n", " Ir = quad(fr, x1, x2)[0] # Perform the inverse Fourier transform integral for the real part\n", " Ii = -quad(fi, x1, x2)[0] # Perform the inverse Fourier transform integral for the imaginary part\n", " psi_tilde[m] = (Ir+1j*Ii)/L**0.5\n", "# print('psi_tilde = ', psi_tilde[m], 'm = ',m-l)\n", "# Reconstruct psi\n", "psi = np.zeros((Nx+1),dtype=complex) # array to hold the complex value of psi(x)\n", "psi_mag = np.zeros((Nx+1)) # array to hold the magnitude of psi(x) \n", "x = np.zeros((Nx+1)) # array to hold the values of the position used for plotting \n", "def wf(l): # define a function that reconstructs the wave function from 2l+1 terms in the Fourier series.\n", " for n in range(0,Nx+1):\n", " x[n]=n*Dx\n", " psi[n] = 0+0j\n", " for m in range(-l,l+1):\n", " psi[n] = psi[n]+psi_tilde[m]*np.e**(+2j*np.pi*(m-l)*x[n]/L)/L**0.5\n", " psi_mag[n] = (psi.real[n]**2+psi.imag[n]**2)**0.5\n", " return psi_mag\n", "MPSI = np.zeros((Nx+1))\n", "fig = plt.figure(figsize=(10,4))\n", "ax = fig.add_subplot(1,1,1)\n", "case=0 # Use this integer below to shift the position of each example to make them easier to compare.\n", "for l in (10,20,40,80,100,200):\n", " MPSI = wf(l) \n", " ax.plot(x+case,MPSI,'r-') #Add a curve described by the arrays x and psi.real to Plot, choose a red solid curve.\n", " case=case+2\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.3" } }, "nbformat": 4, "nbformat_minor": 2 }