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July 14, 2018 05:16
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Hermite polynomials
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| { | |
| "cells": [ | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "using SymPy\nusing Plots; gr(); ENV[\"PLOTS_TEST\"] = \"true\";", | |
| "execution_count": 1, | |
| "outputs": [] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "# Hermite polynomial\nH = sympy[:hermite]\n\n# squared norm of H_n\nc(n) = √π * 2^n * gamma(n+1)\nc(n::Sym) = √Sym(π) * 2^n * gamma(n+1)\n\n# square of pure state\nφ²(n,x) = H(n,x)^2*exp(-x^2)/c(n)\n\n# plot\nf(n,x) = float(φ²(n,x)) # square of pure state\np(r,x) = 1/√(r^2-x^2)/π # arcsine distribution\nPP = []\nfor n in [15, 30, 60]\n r = √(2n+1)\n x = -1.05r:0.05:1.05r\n xx = -0.995r:0.05:0.995r\n y = f.(n,x)\n yy = p.(r,xx)\n ymax = min(yy[1], yy[end])\n P = plot(legend=false, ylim=(0,ymax))\n plot!(title=\"n = $n\", titlefontsize=10)\n plot!(x, y)\n plot!(xx, yy)\n push!(PP, P)\nend\nplot(PP..., size=(750, 180), layout=@layout([a b c]))", | |
| "execution_count": 2, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 2, | |
| "data": { | |
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| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = symbols(\"x\", real=true)\n[H(n,x) for n in 0:10]", | |
| "execution_count": 3, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 3, | |
| "data": { | |
| "text/plain": "11-element Array{SymPy.Sym,1}:\n 1\n 2*x\n 4*x^2 - 2\n 8*x^3 - 12*x\n 16*x^4 - 48*x^2 + 12\n 32*x^5 - 160*x^3 + 120*x\n 64*x^6 - 480*x^4 + 720*x^2 - 120\n 128*x^7 - 1344*x^5 + 3360*x^3 - 1680*x\n 256*x^8 - 3584*x^6 + 13440*x^4 - 13440*x^2 + 1680\n 512*x^9 - 9216*x^7 + 48384*x^5 - 80640*x^3 + 30240*x\n 1024*x^10 - 23040*x^8 + 161280*x^6 - 403200*x^4 + 302400*x^2 - 30240", | |
| "text/latex": "\\begin{bmatrix}1\\\\2 x\\\\4 x^{2} - 2\\\\8 x^{3} - 12 x\\\\16 x^{4} - 48 x^{2} + 12\\\\32 x^{5} - 160 x^{3} + 120 x\\\\64 x^{6} - 480 x^{4} + 720 x^{2} - 120\\\\128 x^{7} - 1344 x^{5} + 3360 x^{3} - 1680 x\\\\256 x^{8} - 3584 x^{6} + 13440 x^{4} - 13440 x^{2} + 1680\\\\512 x^{9} - 9216 x^{7} + 48384 x^{5} - 80640 x^{3} + 30240 x\\\\1024 x^{10} - 23040 x^{8} + 161280 x^{6} - 403200 x^{4} + 302400 x^{2} - 30240\\end{bmatrix}" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = symbols(\"x\", real=true)\nvcat([[integrate(H(n,x)*H(n,x)*exp(-x^2), (x,-oo,oo)) c(Sym(n))] for n in 0:10]...)", | |
| "execution_count": 4, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 4, | |
| "data": { | |
| "text/plain": "11×2 Array{SymPy.Sym,2}:\n sqrt(pi) sqrt(pi)\n 2*sqrt(pi) 2*sqrt(pi)\n 8*sqrt(pi) 8*sqrt(pi)\n 48*sqrt(pi) 48*sqrt(pi)\n 384*sqrt(pi) 384*sqrt(pi)\n 3840*sqrt(pi) 3840*sqrt(pi)\n 46080*sqrt(pi) 46080*sqrt(pi)\n 645120*sqrt(pi) 645120*sqrt(pi)\n 10321920*sqrt(pi) 10321920*sqrt(pi)\n 185794560*sqrt(pi) 185794560*sqrt(pi)\n 3715891200*sqrt(pi) 3715891200*sqrt(pi)", | |
| "text/latex": "\\begin{bmatrix}\\sqrt{\\pi}&\\sqrt{\\pi}\\\\2 \\sqrt{\\pi}&2 \\sqrt{\\pi}\\\\8 \\sqrt{\\pi}&8 \\sqrt{\\pi}\\\\48 \\sqrt{\\pi}&48 \\sqrt{\\pi}\\\\384 \\sqrt{\\pi}&384 \\sqrt{\\pi}\\\\3840 \\sqrt{\\pi}&3840 \\sqrt{\\pi}\\\\46080 \\sqrt{\\pi}&46080 \\sqrt{\\pi}\\\\645120 \\sqrt{\\pi}&645120 \\sqrt{\\pi}\\\\10321920 \\sqrt{\\pi}&10321920 \\sqrt{\\pi}\\\\185794560 \\sqrt{\\pi}&185794560 \\sqrt{\\pi}\\\\3715891200 \\sqrt{\\pi}&3715891200 \\sqrt{\\pi}\\end{bmatrix}" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "", | |
| "execution_count": null, | |
| "outputs": [] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "name": "julia-0.6", | |
| "display_name": "Julia 0.6.4", | |
| "language": "julia" | |
| }, | |
| "toc": { | |
| "nav_menu": {}, | |
| "number_sections": true, | |
| "sideBar": true, | |
| "skip_h1_title": false, | |
| "title_cell": "Table of Contents", | |
| "title_sidebar": "Contents", | |
| "toc_cell": false, | |
| "toc_position": {}, | |
| "toc_section_display": true, | |
| "toc_window_display": false | |
| }, | |
| "language_info": { | |
| "file_extension": ".jl", | |
| "name": "julia", | |
| "mimetype": "application/julia", | |
| "version": "0.6.4" | |
| }, | |
| "gist": { | |
| "id": "", | |
| "data": { | |
| "description": "Hermite polynomials", | |
| "public": true | |
| } | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 2 | |
| } |
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