{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# SageMath para físicos\n", "***\n", "Rogério T. C." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Aula III" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. Vetores e matrizes" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%display latex\n", "reset()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Vetores e estrutura" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vetores são constrídos pelo comando `vector`, com sintaxe `vector([x1,x2,x3,...,xn])`" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "v = vector([x,x^8,5])" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector space of dimension 3 over Symbolic Ring\n" ] } ], "source": [ "print(parent(v))" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Soma, multiplicação escalar e produto vetorial" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "w=vector([5,2,1])" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(x^{8} - 10,\\,-x + 25,\\,-5 \\, x^{8} + 2 \\, x\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(x^{8} - 10,\\,-x + 25,\\,-5 \\, x^{8} + 2 \\, x\\right)\n", "\\end{math}" ], "text/plain": [ "(x^8 - 10, -x + 25, -5*x^8 + 2*x)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.cross_product(w)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Produto interno" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, x^{8} + 5 \\, x + 5$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, x^{8} + 5 \\, x + 5\n", "\\end{math}" ], "text/plain": [ "2*x^8 + 5*x + 5" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v*w" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, x^{8} + 5 \\, x + 5$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, x^{8} + 5 \\, x + 5\n", "\\end{math}" ], "text/plain": [ "2*x^8 + 5*x + 5" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.dot_product(w)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Hermitiano" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}y$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}y\n", "\\end{math}" ], "text/plain": [ "y" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('y', domain='real')" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "z = vector([I*2,x,y]);" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, x + y + 10 i$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, x + y + 10 i\n", "\\end{math}" ], "text/plain": [ "2*x + y + 10*I" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z*w" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}y + 2 \\, \\overline{x} - 10 i$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}y + 2 \\, \\overline{x} - 10 i\n", "\\end{math}" ], "text/plain": [ "y + 2*conjugate(x) - 10*I" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z.hermitian_inner_product(w)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Norma" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{30}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{30}\n", "\\end{math}" ], "text/plain": [ "sqrt(30)" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w.norm()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{y^{2} + {\\left| x \\right|}^{2} + 4}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{y^{2} + {\\left| x \\right|}^{2} + 4}\n", "\\end{math}" ], "text/plain": [ "sqrt(y^2 + abs(x)^2 + 4)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z.norm()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Matrizes" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "1 & 2 & 3 \\\\\n", "5 & 23 & 8 \\\\\n", "8 & 5 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "1 & 2 & 3 \\\\\n", "5 & 23 & 8 \\\\\n", "8 & 5 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 1 2 3]\n", "[ 5 23 8]\n", "[ 8 5 1]" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = matrix(SR,[[1,2,3],[5,23,8],[8,5,1]]);a" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(5 \\, x + 8 \\, y + 2 i,\\,23 \\, x + 5 \\, y + 4 i,\\,8 \\, x + y + 6 i\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(5 \\, x + 8 \\, y + 2 i,\\,23 \\, x + 5 \\, y + 4 i,\\,8 \\, x + y + 6 i\\right)\n", "\\end{math}" ], "text/plain": [ "(5*x + 8*y + 2*I, 23*x + 5*y + 4*I, 8*x + y + 6*I)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(a).transpose()*z" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(5 \\, x + 8 \\, y + 2 i,\\,23 \\, x + 5 \\, y + 4 i,\\,8 \\, x + y + 6 i\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(5 \\, x + 8 \\, y + 2 i,\\,23 \\, x + 5 \\, y + 4 i,\\,8 \\, x + y + 6 i\\right)\n", "\\end{math}" ], "text/plain": [ "(5*x + 8*y + 2*I, 23*x + 5*y + 4*I, 8*x + y + 6*I)" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z*a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Matrizes especiais" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrrrrrrrr}\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrrrrrrrr}\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "matrix.zero(10,15)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[1 0 0 0]\n", "[0 1 0 0]\n", "[0 0 1 0]\n", "[0 0 0 1]" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "matrix.identity(4)" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "1 & 1 & 1 \\\\\n", "1 & 1 & 1 \\\\\n", "1 & 1 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "1 & 1 & 1 \\\\\n", "1 & 1 & 1 \\\\\n", "1 & 1 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[1 1 1]\n", "[1 1 1]\n", "[1 1 1]" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "matrix.ones(3)" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "d = matrix.diagonal([y,y,y^2])" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Determinante, traço, transposta, inversa, conjugada" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{4}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{4}\n", "\\end{math}" ], "text/plain": [ "y^4" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d.det()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{2} + 2 \\, y$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{2} + 2 \\, y\n", "\\end{math}" ], "text/plain": [ "y^2 + 2*y" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d.trace()" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "y & 0 & 0 \\\\\n", "0 & y & 0 \\\\\n", "0 & 0 & y^{2}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "y & 0 & 0 \\\\\n", "0 & y & 0 \\\\\n", "0 & 0 & y^{2}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ y 0 0]\n", "[ 0 y 0]\n", "[ 0 0 y^2]" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d.transpose()" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "y & 0 & 0 \\\\\n", "0 & y & 0 \\\\\n", "0 & 0 & y^{2}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "y & 0 & 0 \\\\\n", "0 & y & 0 \\\\\n", "0 & 0 & y^{2}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ y 0 0]\n", "[ 0 y 0]\n", "[ 0 0 y^2]" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d.conjugate()" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "\\frac{1}{y} & 0 & 0 \\\\\n", "0 & \\frac{1}{y} & 0 \\\\\n", "0 & 0 & \\frac{1}{y^{2}}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "\\frac{1}{y} & 0 & 0 \\\\\n", "0 & \\frac{1}{y} & 0 \\\\\n", "0 & 0 & \\frac{1}{y^{2}}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 1/y 0 0]\n", "[ 0 1/y 0]\n", "[ 0 0 y^(-2)]" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d.inverse()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Pausa - transformação linear" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [], "source": [ "T = linear_transformation(a)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\text{vector space morphism from }\n", "\\text{SR}^{3}\\text{ to }\n", "\\text{SR}^{3}\\text{ represented by the matrix }\n", "\\left(\\begin{array}{rrr}\n", "1 & 2 & 3 \\\\\n", "5 & 23 & 8 \\\\\n", "8 & 5 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\text{vector space morphism from }\n", "\\text{SR}^{3}\\text{ to }\n", "\\text{SR}^{3}\\text{ represented by the matrix }\n", "\\left(\\begin{array}{rrr}\n", "1 & 2 & 3 \\\\\n", "5 & 23 & 8 \\\\\n", "8 & 5 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "Vector space morphism represented by the matrix:\n", "[ 1 2 3]\n", "[ 5 23 8]\n", "[ 8 5 1]\n", "Domain: Vector space of dimension 3 over Symbolic Ring\n", "Codomain: Vector space of dimension 3 over Symbolic Ring" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector space morphism represented by the matrix:\n", "[ 1 2 3]\n", "[ 5 23 8]\n", "[ 8 5 1]\n", "Domain: Vector space of dimension 3 over Symbolic Ring\n", "Codomain: Vector space of dimension 3 over Symbolic Ring\n" ] } ], "source": [ "print(T)" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T.is_bijective()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Autovalores, auto vetores e diagonalização" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{3} + \\left(-y^{2} - 2 \\, y\\right) x^{2} + \\left(2 \\, y^{3} + y^{2}\\right) x - y^{4}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{3} + \\left(-y^{2} - 2 \\, y\\right) x^{2} + \\left(2 \\, y^{3} + y^{2}\\right) x - y^{4}\n", "\\end{math}" ], "text/plain": [ "x^3 + (-y^2 - 2*y)*x^2 + (2*y^3 + y^2)*x - y^4" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d.charpoly()" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-\\frac{1}{2} \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} {\\left(i \\, \\sqrt{3} + 1\\right)} - \\frac{353 \\, {\\left(-i \\, \\sqrt{3} + 1\\right)}}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, -\\frac{1}{2} \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} {\\left(-i \\, \\sqrt{3} + 1\\right)} - \\frac{353 \\, {\\left(i \\, \\sqrt{3} + 1\\right)}}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} + \\frac{706}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-\\frac{1}{2} \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} {\\left(i \\, \\sqrt{3} + 1\\right)} - \\frac{353 \\, {\\left(-i \\, \\sqrt{3} + 1\\right)}}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, -\\frac{1}{2} \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} {\\left(-i \\, \\sqrt{3} + 1\\right)} - \\frac{353 \\, {\\left(i \\, \\sqrt{3} + 1\\right)}}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} + \\frac{706}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}\\right]\n", "\\end{math}" ], "text/plain": [ "[-1/2*(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3)*(I*sqrt(3) + 1) - 353/9*(-I*sqrt(3) + 1)/(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3) + 25/3,\n", " -1/2*(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3)*(-I*sqrt(3) + 1) - 353/9*(I*sqrt(3) + 1)/(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3) + 25/3,\n", " (1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3) + 706/9/(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3) + 25/3]" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.eigenvalues()" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(-\\frac{1}{2} \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} {\\left(i \\, \\sqrt{3} + 1\\right)} - \\frac{353 \\, {\\left(-i \\, \\sqrt{3} + 1\\right)}}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, \\left[\\right], 1\\right), \\left(-\\frac{1}{2} \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} {\\left(-i \\, \\sqrt{3} + 1\\right)} - \\frac{353 \\, {\\left(i \\, \\sqrt{3} + 1\\right)}}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, \\left[\\right], 1\\right), \\left({\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} + \\frac{706}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, \\left[\\right], 1\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(-\\frac{1}{2} \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} {\\left(i \\, \\sqrt{3} + 1\\right)} - \\frac{353 \\, {\\left(-i \\, \\sqrt{3} + 1\\right)}}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, \\left[\\right], 1\\right), \\left(-\\frac{1}{2} \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} {\\left(-i \\, \\sqrt{3} + 1\\right)} - \\frac{353 \\, {\\left(i \\, \\sqrt{3} + 1\\right)}}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, \\left[\\right], 1\\right), \\left({\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}} + \\frac{706}{9 \\, {\\left(\\frac{1}{18} i \\, \\sqrt{24785605} \\sqrt{3} + \\frac{27173}{54}\\right)}^{\\frac{1}{3}}} + \\frac{25}{3}, \\left[\\right], 1\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[(-1/2*(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3)*(I*sqrt(3) + 1) - 353/9*(-I*sqrt(3) + 1)/(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3) + 25/3,\n", " [],\n", " 1),\n", " (-1/2*(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3)*(-I*sqrt(3) + 1) - 353/9*(I*sqrt(3) + 1)/(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3) + 25/3,\n", " [],\n", " 1),\n", " ((1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3) + 706/9/(1/18*I*sqrt(24785605)*sqrt(3) + 27173/54)^(1/3) + 25/3,\n", " [],\n", " 1)]" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.eigenvectors_right()" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [], "source": [ "k = matrix([[1,1,-3],[0,4,0],[-3,3,1]])" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(-2, \\left[\\left(1,\\,0,\\,1\\right)\\right], 1\\right), \\left(4, \\left[\\left(1,\\,0,\\,-1\\right)\\right], 2\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(-2, \\left[\\left(1,\\,0,\\,1\\right)\\right], 1\\right), \\left(4, \\left[\\left(1,\\,0,\\,-1\\right)\\right], 2\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[(-2,\n", " [\n", " (1, 0, 1)\n", " ],\n", " 1),\n", " (4,\n", " [\n", " (1, 0, -1)\n", " ],\n", " 2)]" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.eigenvectors_right()" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\left(\\begin{array}{rrr}\n", "-2 & 0 & 0 \\\\\n", "0 & 4 & 0 \\\\\n", "0 & 0 & 4\n", "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", "1 & 1 & 0 \\\\\n", "0 & 0 & 0 \\\\\n", "1 & -1 & 0\n", "\\end{array}\\right)\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\left(\\begin{array}{rrr}\n", "-2 & 0 & 0 \\\\\n", "0 & 4 & 0 \\\\\n", "0 & 0 & 4\n", "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", "1 & 1 & 0 \\\\\n", "0 & 0 & 0 \\\\\n", "1 & -1 & 0\n", "\\end{array}\\right)\\right)\n", "\\end{math}" ], "text/plain": [ "(\n", "[-2 0 0] [ 1 1 0]\n", "[ 0 4 0] [ 0 0 0]\n", "[ 0 0 4], [ 1 -1 0]\n", ")" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.eigenmatrix_right()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Exponencial de matrizes" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$e^A \\equiv I+\\sum_{n=1}^{\\infty}\\frac{A^n}{n!},$$\n", "onde ${\\displaystyle I\\,}$ é a matriz identidade" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "1 & 2 \\\\\n", "2 & 2\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "1 & 2 \\\\\n", "2 & 2\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[1 2]\n", "[2 2]" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m1 = matrix([[1,2],[2,2]]);m1" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "-\\frac{1}{34} \\, {\\left({\\left(\\sqrt{17} - 17\\right)} e^{\\sqrt{17}} - \\sqrt{17} - 17\\right)} e^{\\left(-\\frac{1}{2} \\, \\sqrt{17} + \\frac{3}{2}\\right)} & \\frac{2}{17} \\, {\\left(\\sqrt{17} e^{\\sqrt{17}} - \\sqrt{17}\\right)} e^{\\left(-\\frac{1}{2} \\, \\sqrt{17} + \\frac{3}{2}\\right)} \\\\\n", "\\frac{2}{17} \\, {\\left(\\sqrt{17} e^{\\sqrt{17}} - \\sqrt{17}\\right)} e^{\\left(-\\frac{1}{2} \\, \\sqrt{17} + \\frac{3}{2}\\right)} & \\frac{1}{34} \\, {\\left({\\left(\\sqrt{17} + 17\\right)} e^{\\sqrt{17}} - \\sqrt{17} + 17\\right)} e^{\\left(-\\frac{1}{2} \\, \\sqrt{17} + \\frac{3}{2}\\right)}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "-\\frac{1}{34} \\, {\\left({\\left(\\sqrt{17} - 17\\right)} e^{\\sqrt{17}} - \\sqrt{17} - 17\\right)} e^{\\left(-\\frac{1}{2} \\, \\sqrt{17} + \\frac{3}{2}\\right)} & \\frac{2}{17} \\, {\\left(\\sqrt{17} e^{\\sqrt{17}} - \\sqrt{17}\\right)} e^{\\left(-\\frac{1}{2} \\, \\sqrt{17} + \\frac{3}{2}\\right)} \\\\\n", "\\frac{2}{17} \\, {\\left(\\sqrt{17} e^{\\sqrt{17}} - \\sqrt{17}\\right)} e^{\\left(-\\frac{1}{2} \\, \\sqrt{17} + \\frac{3}{2}\\right)} & \\frac{1}{34} \\, {\\left({\\left(\\sqrt{17} + 17\\right)} e^{\\sqrt{17}} - \\sqrt{17} + 17\\right)} e^{\\left(-\\frac{1}{2} \\, \\sqrt{17} + \\frac{3}{2}\\right)}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[-1/34*((sqrt(17) - 17)*e^sqrt(17) - sqrt(17) - 17)*e^(-1/2*sqrt(17) + 3/2) 2/17*(sqrt(17)*e^sqrt(17) - sqrt(17))*e^(-1/2*sqrt(17) + 3/2)]\n", "[ 2/17*(sqrt(17)*e^sqrt(17) - sqrt(17))*e^(-1/2*sqrt(17) + 3/2) 1/34*((sqrt(17) + 17)*e^sqrt(17) - sqrt(17) + 17)*e^(-1/2*sqrt(17) + 3/2)]" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "exp(m1)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Exercício\n", "\n", "Dada a matriz de Pauli\n", "$${\\displaystyle \\sigma _{2}={\\begin{pmatrix}0&i\\\\-i&0\\end{pmatrix}}.}$$\n", "\n", "Calcule $e^{i\\frac{\\theta}{2}\\sigma_2}$" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\theta$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\theta\n", "\\end{math}" ], "text/plain": [ "theta" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('theta')" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "\\frac{1}{2} \\, {\\left(e^{\\left(i \\, \\theta\\right)} + 1\\right)} e^{\\left(-\\frac{1}{2} i \\, \\theta\\right)} & \\frac{1}{2} \\, {\\left(i \\, e^{\\left(i \\, \\theta\\right)} - i\\right)} e^{\\left(-\\frac{1}{2} i \\, \\theta\\right)} \\\\\n", "-\\frac{1}{2} \\, {\\left(i \\, e^{\\left(i \\, \\theta\\right)} - i\\right)} e^{\\left(-\\frac{1}{2} i \\, \\theta\\right)} & \\frac{1}{2} \\, {\\left(e^{\\left(i \\, \\theta\\right)} + 1\\right)} e^{\\left(-\\frac{1}{2} i \\, \\theta\\right)}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "\\frac{1}{2} \\, {\\left(e^{\\left(i \\, \\theta\\right)} + 1\\right)} e^{\\left(-\\frac{1}{2} i \\, \\theta\\right)} & \\frac{1}{2} \\, {\\left(i \\, e^{\\left(i \\, \\theta\\right)} - i\\right)} e^{\\left(-\\frac{1}{2} i \\, \\theta\\right)} \\\\\n", "-\\frac{1}{2} \\, {\\left(i \\, e^{\\left(i \\, \\theta\\right)} - i\\right)} e^{\\left(-\\frac{1}{2} i \\, \\theta\\right)} & \\frac{1}{2} \\, {\\left(e^{\\left(i \\, \\theta\\right)} + 1\\right)} e^{\\left(-\\frac{1}{2} i \\, \\theta\\right)}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 1/2*(e^(I*theta) + 1)*e^(-1/2*I*theta) 1/2*(I*e^(I*theta) - I)*e^(-1/2*I*theta)]\n", "[-1/2*(I*e^(I*theta) - I)*e^(-1/2*I*theta) 1/2*(e^(I*theta) + 1)*e^(-1/2*I*theta)]" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "exp((i/2)*theta*matrix([[0,i],[-i,0]]))" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}x$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}x\n", "\\end{math}" ], "text/plain": [ "x" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x.simplify_rectform()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Geradores do grupo de Lorentz (Representação de spin 1 da algébra de Lie $\\mathfrak{so}(1,3)$)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Rotações" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [], "source": [ "J1 = I*matrix([[0,0,0,0],[0,0,0,0],[0,0,0,-1],[0,0,1,0]])\n", "J2 = I*matrix([[0,0,0,0],[0,0,0,1],[0,0,0,0],[0,-1,0,0]])\n", "J3 = I*matrix([[0,0,0,0],[0,0,-1,0],[0,1,0,0],[0,0,0,0]])" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & -i \\\\\n", "0 & 0 & i & 0\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & i \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & -i & 0 & 0\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & -i & 0 \\\\\n", "0 & i & 0 & 0 \\\\\n", "0 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & -i \\\\\n", "0 & 0 & i & 0\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & i \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & -i & 0 & 0\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & -i & 0 \\\\\n", "0 & i & 0 & 0 \\\\\n", "0 & 0 & 0 & 0\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 0 0 0 0]\n", "[ 0 0 0 0]\n", "[ 0 0 0 -I]\n", "[ 0 0 I 0] [ 0 0 0 0]\n", "[ 0 0 0 I]\n", "[ 0 0 0 0]\n", "[ 0 -I 0 0] [ 0 0 0 0]\n", "[ 0 0 -I 0]\n", "[ 0 I 0 0]\n", "[ 0 0 0 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(J1,J2,J3)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\theta_{0}, \\theta_{1}, \\theta_{2}, \\theta_{3}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\theta_{0}, \\theta_{1}, \\theta_{2}, \\theta_{3}\\right)\n", "\\end{math}" ], "text/plain": [ "(theta0, theta1, theta2, theta3)" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "th = var('theta',n=4);th" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [], "source": [ "R1 = exp(I*th[1]*J1)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [], "source": [ "R2 = exp(I*th[2]*J2)" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "scrolled": true }, "outputs": [], "source": [ "R3 = exp(I*th[3]*J3)" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{1}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{1}\\right)} & -\\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{1}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{1}\\right)} \\\\\n", "0 & 0 & \\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{1}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{1}\\right)} & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{1}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{1}\\right)}\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{2}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{2}\\right)} & 0 & \\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{2}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{2}\\right)} \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & -\\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{2}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{2}\\right)} & 0 & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{2}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{2}\\right)}\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{3}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{3}\\right)} & -\\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{3}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{3}\\right)} & 0 \\\\\n", "0 & \\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{3}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{3}\\right)} & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{3}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{3}\\right)} & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{1}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{1}\\right)} & -\\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{1}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{1}\\right)} \\\\\n", "0 & 0 & \\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{1}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{1}\\right)} & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{1}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{1}\\right)}\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{2}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{2}\\right)} & 0 & \\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{2}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{2}\\right)} \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & -\\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{2}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{2}\\right)} & 0 & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{2}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{2}\\right)}\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{3}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{3}\\right)} & -\\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{3}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{3}\\right)} & 0 \\\\\n", "0 & \\frac{1}{2} \\, {\\left(i \\, e^{\\left(2 i \\, \\theta_{3}\\right)} - i\\right)} e^{\\left(-i \\, \\theta_{3}\\right)} & \\frac{1}{2} \\, {\\left(e^{\\left(2 i \\, \\theta_{3}\\right)} + 1\\right)} e^{\\left(-i \\, \\theta_{3}\\right)} & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 1 0 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 1/2*(e^(2*I*theta1) + 1)*e^(-I*theta1) -1/2*(I*e^(2*I*theta1) - I)*e^(-I*theta1)]\n", "[ 0 0 1/2*(I*e^(2*I*theta1) - I)*e^(-I*theta1) 1/2*(e^(2*I*theta1) + 1)*e^(-I*theta1)] [ 1 0 0 0]\n", "[ 0 1/2*(e^(2*I*theta2) + 1)*e^(-I*theta2) 0 1/2*(I*e^(2*I*theta2) - I)*e^(-I*theta2)]\n", "[ 0 0 1 0]\n", "[ 0 -1/2*(I*e^(2*I*theta2) - I)*e^(-I*theta2) 0 1/2*(e^(2*I*theta2) + 1)*e^(-I*theta2)] [ 1 0 0 0]\n", "[ 0 1/2*(e^(2*I*theta3) + 1)*e^(-I*theta3) -1/2*(I*e^(2*I*theta3) - I)*e^(-I*theta3) 0]\n", "[ 0 1/2*(I*e^(2*I*theta3) - I)*e^(-I*theta3) 1/2*(e^(2*I*theta3) + 1)*e^(-I*theta3) 0]\n", "[ 0 0 0 1]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(R1,R2,R3)" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\cos\\left(\\theta_{1}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\cos\\left(\\theta_{1}\\right)\n", "\\end{math}" ], "text/plain": [ "cos(theta1)" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R1[2,2].expand().simplify_rectform()" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & \\frac{1}{2} \\, e^{\\left(i \\, \\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\left(-i \\, \\theta_{1}\\right)} & -\\frac{1}{2} i \\, e^{\\left(i \\, \\theta_{1}\\right)} + \\frac{1}{2} i \\, e^{\\left(-i \\, \\theta_{1}\\right)} \\\\\n", "0 & 0 & \\frac{1}{2} i \\, e^{\\left(i \\, \\theta_{1}\\right)} - \\frac{1}{2} i \\, e^{\\left(-i \\, \\theta_{1}\\right)} & \\frac{1}{2} \\, e^{\\left(i \\, \\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\left(-i \\, \\theta_{1}\\right)}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & \\frac{1}{2} \\, e^{\\left(i \\, \\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\left(-i \\, \\theta_{1}\\right)} & -\\frac{1}{2} i \\, e^{\\left(i \\, \\theta_{1}\\right)} + \\frac{1}{2} i \\, e^{\\left(-i \\, \\theta_{1}\\right)} \\\\\n", "0 & 0 & \\frac{1}{2} i \\, e^{\\left(i \\, \\theta_{1}\\right)} - \\frac{1}{2} i \\, e^{\\left(-i \\, \\theta_{1}\\right)} & \\frac{1}{2} \\, e^{\\left(i \\, \\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\left(-i \\, \\theta_{1}\\right)}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 1 0 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 1/2*e^(I*theta1) + 1/2*e^(-I*theta1) -1/2*I*e^(I*theta1) + 1/2*I*e^(-I*theta1)]\n", "[ 0 0 1/2*I*e^(I*theta1) - 1/2*I*e^(-I*theta1) 1/2*e^(I*theta1) + 1/2*e^(-I*theta1)]" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R1.expand()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Funçãozinha em Python" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [], "source": [ "def resolve(M):\n", " for lin in range(4):\n", " for col in range(4):\n", " M[lin,col] = M[lin,col].expand().simplify_rectform()\n", " return M" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\cos\\left(\\theta_{3}\\right) & \\sin\\left(\\theta_{3}\\right) & 0 \\\\\n", "0 & -\\sin\\left(\\theta_{3}\\right) & \\cos\\left(\\theta_{3}\\right) & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\cos\\left(\\theta_{3}\\right) & \\sin\\left(\\theta_{3}\\right) & 0 \\\\\n", "0 & -\\sin\\left(\\theta_{3}\\right) & \\cos\\left(\\theta_{3}\\right) & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 1 0 0 0]\n", "[ 0 cos(theta3) sin(theta3) 0]\n", "[ 0 -sin(theta3) cos(theta3) 0]\n", "[ 0 0 0 1]" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "resolve(R1);resolve(R2);resolve(R3)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & \\cos\\left(\\theta_{1}\\right) & \\sin\\left(\\theta_{1}\\right) \\\\\n", "0 & 0 & -\\sin\\left(\\theta_{1}\\right) & \\cos\\left(\\theta_{1}\\right)\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\cos\\left(\\theta_{2}\\right) & 0 & -\\sin\\left(\\theta_{2}\\right) \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & \\sin\\left(\\theta_{2}\\right) & 0 & \\cos\\left(\\theta_{2}\\right)\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\cos\\left(\\theta_{3}\\right) & \\sin\\left(\\theta_{3}\\right) & 0 \\\\\n", "0 & -\\sin\\left(\\theta_{3}\\right) & \\cos\\left(\\theta_{3}\\right) & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & \\cos\\left(\\theta_{1}\\right) & \\sin\\left(\\theta_{1}\\right) \\\\\n", "0 & 0 & -\\sin\\left(\\theta_{1}\\right) & \\cos\\left(\\theta_{1}\\right)\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\cos\\left(\\theta_{2}\\right) & 0 & -\\sin\\left(\\theta_{2}\\right) \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & \\sin\\left(\\theta_{2}\\right) & 0 & \\cos\\left(\\theta_{2}\\right)\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & \\cos\\left(\\theta_{3}\\right) & \\sin\\left(\\theta_{3}\\right) & 0 \\\\\n", "0 & -\\sin\\left(\\theta_{3}\\right) & \\cos\\left(\\theta_{3}\\right) & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 1 0 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 cos(theta1) sin(theta1)]\n", "[ 0 0 -sin(theta1) cos(theta1)] [ 1 0 0 0]\n", "[ 0 cos(theta2) 0 -sin(theta2)]\n", "[ 0 0 1 0]\n", "[ 0 sin(theta2) 0 cos(theta2)] [ 1 0 0 0]\n", "[ 0 cos(theta3) sin(theta3) 0]\n", "[ 0 -sin(theta3) cos(theta3) 0]\n", "[ 0 0 0 1]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(R1,R2,R3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Boosts" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [], "source": [ "K1 = -I*matrix([[0,1,0,0],[1,0,0,0],[0,0,0,0],[0,0,0,0]])\n", "K2 = -I*matrix([[0,0,1,0],[0,0,0,0],[1,0,0,0],[0,0,0,0]])\n", "K3 = -I*matrix([[0,0,0,1],[0,0,0,0],[0,0,0,0],[1,0,0,0]])" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "0 & -i & 0 & 0 \\\\\n", "-i & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "0 & 0 & -i & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "-i & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & -i \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "-i & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "0 & -i & 0 & 0 \\\\\n", "-i & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "0 & 0 & -i & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "-i & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & -i \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "-i & 0 & 0 & 0\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 0 -I 0 0]\n", "[-I 0 0 0]\n", "[ 0 0 0 0]\n", "[ 0 0 0 0] [ 0 0 -I 0]\n", "[ 0 0 0 0]\n", "[-I 0 0 0]\n", "[ 0 0 0 0] [ 0 0 0 -I]\n", "[ 0 0 0 0]\n", "[ 0 0 0 0]\n", "[-I 0 0 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(K1,K2,K3)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [], "source": [ "B1 = exp(I*th[1]*K1).expand()\n", "B2 = exp(I*th[1]*K2).expand()\n", "B3 = exp(I*th[1]*K3).expand()" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & -\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & 0 \\\\\n", "-\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & \\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & -\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "-\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & \\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & 0 & -\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "-\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & 0 & \\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & -\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & 0 \\\\\n", "-\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & \\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & -\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "-\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & \\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right) \\left(\\begin{array}{rrrr}\n", "\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & 0 & -\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "-\\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}} & 0 & 0 & \\frac{1}{2} \\, e^{\\left(-\\theta_{1}\\right)} + \\frac{1}{2} \\, e^{\\theta_{1}}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 1/2*e^(-theta1) + 1/2*e^theta1 -1/2*e^(-theta1) + 1/2*e^theta1 0 0]\n", "[-1/2*e^(-theta1) + 1/2*e^theta1 1/2*e^(-theta1) + 1/2*e^theta1 0 0]\n", "[ 0 0 1 0]\n", "[ 0 0 0 1] [ 1/2*e^(-theta1) + 1/2*e^theta1 0 -1/2*e^(-theta1) + 1/2*e^theta1 0]\n", "[ 0 1 0 0]\n", "[-1/2*e^(-theta1) + 1/2*e^theta1 0 1/2*e^(-theta1) + 1/2*e^theta1 0]\n", "[ 0 0 0 1] [ 1/2*e^(-theta1) + 1/2*e^theta1 0 0 -1/2*e^(-theta1) + 1/2*e^theta1]\n", "[ 0 1 0 0]\n", "[ 0 0 1 0]\n", "[-1/2*e^(-theta1) + 1/2*e^theta1 0 0 1/2*e^(-theta1) + 1/2*e^theta1]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(B1,B2,B3)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Verificando que $R_i$ e $B_i$ são transformações de Lorentz\n", "$$\\Lambda^T\\eta\\Lambda = \\eta$$" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[-1 0 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 1 0]\n", "[ 0 0 0 1]" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eta = diagonal_matrix([-1,1,1,1]);eta" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[-1 0 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 1 0]\n", "[ 0 0 0 1]" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(R1.transpose()*eta*R1).simplify_trig()" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[-1 0 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 1 0]\n", "[ 0 0 0 1]" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(B3.transpose()*eta*B3).expand()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Espaço de Minkowski" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [], "source": [ "M = VectorSpace(SR,4, inner_product_matrix=eta)" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(3,\\,2,\\,1,\\,2\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(3,\\,2,\\,1,\\,2\\right)\n", "\\end{math}" ], "text/plain": [ "(3, 2, 1, 2)" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w = M([3,2,1,2]);w" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(3,\\,0,\\,1,\\,2\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(3,\\,0,\\,1,\\,2\\right)\n", "\\end{math}" ], "text/plain": [ "(3, 0, 1, 2)" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u = M([3,0,1,2]);u" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}18$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}18\n", "\\end{math}" ], "text/plain": [ "18" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w.dot_product(w) #Euclidiano" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$w^\\mu u_\\mu = \\eta_{\\mu\\nu}w^\\mu u^\\nu = \\eta(w,u)$$" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4\n", "\\end{math}" ], "text/plain": [ "-4" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w.inner_product(u) #Minkowski" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4\n", "\\end{math}" ], "text/plain": [ "-4" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w*eta*u" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}0\n", "\\end{math}" ], "text/plain": [ "0" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(w).inner_product(w)" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4\n", "\\end{math}" ], "text/plain": [ "-4" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(u).inner_product(u)" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, \\sqrt{2}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, \\sqrt{2}\n", "\\end{math}" ], "text/plain": [ "3*sqrt(2)" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w.norm()" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 i$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 i\n", "\\end{math}" ], "text/plain": [ "2*I" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sqrt(u*eta*u)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.2", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }