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Created July 12, 2015 03:22
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cpchung
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Derivation for the inner product of 3D rotational gradient operator acted on spherical harmonics "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"with application in object/scenes response to a lighting environment, often represented using spherical harmonics,including complex global illumination effects"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The first paper that\n",
"has been used extensively in games deals with using Spherical Harmonics to represent\n",
"irradiance environment maps efficiently, allowing for interactive rendering of diffuse objects\n",
"under distant illumination"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" \n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Reference about spherical harmonics\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"http://mathworld.wolfram.com/SphericalHarmonic.html"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#Spherical harmonics in Sympy"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To get back the well known expressions in spherical coordinates we use full expansion:"
]
},
{
"cell_type": "code",
"execution_count": 114,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\frac{P_{n}^{\\left(m\\right)}\\left(\\cos{\\left (\\theta \\right )}\\right)}{2 \\sqrt{\\pi}} \\sqrt{\\frac{\\left(- m + n\\right)!}{\\left(m + n\\right)!} \\left(2 n + 1\\right)} e^{i m \\phi}$$"
],
"text/plain": [
" _____________________ \n",
" ╱ (2⋅n + 1)⋅(-m + n)! ⅈ⋅m⋅φ \n",
" ╱ ─────────────────── ⋅ℯ ⋅assoc_legendre(n, m, cos(θ))\n",
"╲╱ (m + n)! \n",
"─────────────────────────────────────────────────────────────\n",
" ___ \n",
" 2⋅╲╱ π "
]
},
"execution_count": 114,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from sympy import Ynm, Symbol, expand_func\n",
"from sympy.abc import n,m\n",
"theta = Symbol(\"theta\")\n",
"phi = Symbol(\"phi\")\n",
"expand_func(Ynm(n, m, theta, phi))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Relevant page\n",
"http://docs.sympy.org/latest/modules/functions/special.html?highlight=spherical%20harmonics#sympy.functions.special.spherical_harmonics.Ynm"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"3d plot of spherical harmonics\n",
"\n",
"http://stsdas.stsci.edu/download/mdroe/plotting/entry1/index.html"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# Derivation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We try to derive the dot product of rotational gradient of spherical harmonics with spherical harmonics expansion."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"$$\\vec{R} = \\begin{bmatrix}\n",
" R_{x}\\\\\n",
" R_{y} \\\\\n",
" R_{z}\n",
" \\end{bmatrix}\n",
" =\\begin{bmatrix}\n",
" - \\frac{cos\\theta}{sin\\theta}cos\\phi \\frac{\\partial }{\\partial \\phi} - sin\\phi \\frac{\\partial }{\\partial \\theta}\n",
" \\\\\n",
" - \\frac{cos\\theta}{sin\\theta}cos\\phi \\frac{\\partial }{\\partial \\phi} + cos\\phi \\frac{\\partial }{\\partial \\theta} \n",
" \\\\\n",
" \\frac{\\partial }{\\partial \\phi} \n",
" \\end{bmatrix}\n",
" $$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"R_{\\alpha} = i L_{\\alpha}, where \\, \\alpha = x,y,z\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" $$ \\vec{R} = \\begin{bmatrix}\n",
" R_{x} \\\\\n",
" R_{y} \\\\\n",
" R_{z}\n",
" \\end{bmatrix}\n",
" =\n",
" \\begin{bmatrix}\n",
" i L_{x} \\\\\n",
" i L_{y} \\\\\n",
" i L_{z}\n",
" \\end{bmatrix}\n",
" =\n",
" \\begin{bmatrix}\n",
" \\frac{i}{2}(L_{+}+L_{-} ) \\\\\n",
" \\frac{1}{2}(L_{+}-L_{-} ) \\\\\n",
" i L_{z}\n",
" \\end{bmatrix}\n",
" $$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"L_{+} Y{_l^{m}}= c_{l,m,1} Y{_l^{m+1}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"$$\n",
"L_{-} Y{_l^{m}}= c_{l,m,2} Y{_l^{m-1}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"$$\n",
"L_{z} Y{_l^{m}}= m Y{_l^{m}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"after some manipulations of the above identities, it can be shown that (details will be added later)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$ \\vec{R}Y{_j^{p}} \\cdot \\vec{R}Y{_l^{m}} = - \\frac{1}{2} [c_{j,p,1}c_{l,m,2}Y{_j^{p+1}} Y{_l^{m-1}}+c_{j,p,2}c_{l,m,1}Y{_j^{p-1}}Y{_l^{m+1}}] - mp Y{_j^{p}} Y{_l^{m}} $$ "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$ =(-1)*(-1)^{m+p} \\sum\\limits_{k=|l-j|}^{l+j}Y{_k^{m+p}}[\\frac{1}{2} (c_{j,p,1}c_{l,m,2} \\alpha_{l,m-1,j,p+1,k} +c_{j,p,2}C_{l,m,1}\\alpha_{l,m+1,j,p-1,k}) + mp \\alpha_{l,m,j,p,k}] $$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"where $$ \\alpha_{l,m,j,p,k} = \\int_{0}^{2\\pi} \\int_{0}^{\\pi} Y{_j^{p}} Y{_l^{m}} Y{_k^{-m-p}} \\sin \\theta d\\theta d\\phi$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$ =[\\frac{(2l+1)(2j+1)(2k+1)}{4\\pi}]^{1/2} * \\left(\\begin{array}{clcr} j & l & k\\\\ 0& 0 & 0 \\end{array}\\right) * \\left(\\begin{array}{clcr} j & l & k\\\\ p& m & -(m+p) \\end{array}\\right) $$ "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"here \n",
"$$\n",
"\\left(\\begin{array}{clcr} j & l & k\\\\ 0& 0 & 0 \\end{array}\\right)\n",
"$$\n",
"\n",
"is Wigner 3j symbol"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Wigner 3j symbol"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"reference for Wigner 3j symbol(strange fraction here)\n",
"\n",
"http://docs.sympy.org/dev/modules/physics/wigner.html"
]
},
{
"cell_type": "code",
"execution_count": 115,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\frac{\\sqrt{715}}{143}$$"
],
"text/plain": [
" _____\n",
"╲╱ 715 \n",
"───────\n",
" 143 "
]
},
"execution_count": 115,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from sympy.physics.wigner import wigner_3j\n",
"wigner_3j(2, 6, 4, 0, 0, 0)"
]
},
{
"cell_type": "code",
"execution_count": 164,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\frac{\\sqrt{715}}{143}$$"
],
"text/plain": [
" _____\n",
"╲╱ 715 \n",
"───────\n",
" 143 "
]
},
"execution_count": 164,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sqrt(S(5)/143)"
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$0$$"
],
"text/plain": [
"0"
]
},
"execution_count": 116,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"wigner_3j(2, 6, 4, 0, 0, 1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"and$$\n",
" Y{_j^{p}} = Y{_j^{p}}(\\theta, \\phi)\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"c_{l,m,1}=\\sqrt{(l-m)(l+m+1)}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"c_{l,m,2}=\\sqrt{(l+m)(l-m+1)}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" using the recursive relation shown p760,34.3.14,NIST Handbook 2010 ,we have:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" $$ \\frac{1}{2} (c_{j,p,1}c_{l,m,2} \\alpha_{l,m-1,j,p+1,k} +c_{j,p,2}C_{l,m,1}\\alpha_{l,m+1,j,p-1,k}) + mp \\alpha_{l,m,j,p,k} $$ "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$ =\\frac{1}{2}[\\frac{(2l+1)(2j+1)(2k+1)}{4\\pi}]^{1/2} * \\left(\\begin{array}{clcr} j & l & k\\\\ 0& 0 & 0 \\end{array}\\right) * [ c_{j,p,1}c_{l,m,2} \\left(\\begin{array}{clcr} j & l & k\\\\ p+1& m-1 & -(m+p) \\end{array}\\right) + c_{j,p,2}c_{l,m,1} \\left(\\begin{array}{clcr} j & l & k\\\\ p-1& m+1 & -(m+p) \\end{array}\\right) + 2mp \\left(\\begin{array}{clcr} j & l & k\\\\ p-1& m+1 & -(m+p) \\end{array}\\right) ] $$ "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$ = \\frac{1}{2}[\\frac{(2l+1)(2j+1)(2k+1)}{4\\pi}]^{1/2} * \\left(\\begin{array}{clcr} j & l & k\\\\ 0& 0 & 0 \\end{array}\\right) * \\left(\\begin{array}{clcr} k & j & l\\\\ -(m+p)& p & m \\end{array}\\right) *(k^2-j^2-l^2+k-j-(-2pm+2pm)) $$ "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" $$ = \\frac{1}{2}[\\frac{(2l+1)(2j+1)(2k+1)}{4\\pi}]^{1/2} * \\left(\\begin{array}{clcr} k & j & l\\\\ 0& 0 & 0 \\end{array}\\right) * \\left(\\begin{array}{clcr} k & j & l\\\\ -(m+p)& p & m \\end{array}\\right) *(k^2-j^2-l^2+k-j-l) $$ "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$ = \\alpha_{l,m,j,p,k} * \\frac{1}{2} (k^2-j^2-l^2+k-j-l) $$ "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## so, we have \n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$ \\vec{R}Y{_j^{p}} \\cdot \\vec{R}Y{_l^{m}} = (-1)^{m+p} \\sum\\limits_{k=|l-j|}^{l+j}Y{_k^{m+p}} * \\alpha_{l,m,j,p,k} * \\frac{1}{2} (k^2-j^2-l^2+k-j-l) $$"
]
},
{
"cell_type": "code",
"execution_count": 161,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0\n"
]
}
],
"source": [
"from sympy import *\n",
"from sympy.physics.wigner import wigner_3j\n",
"init_printing(use_latex='mathjax')\n",
"\n",
"def dot_rota_grad_SH(j, p, l, m, theta, phi):\n",
" temp=0\n",
" for k in range(abs(l-j), l+j +1):\n",
" temp+=Ynm(k, m+p, theta,phi)*alpha(l,m,j,p,k)/2*(k**2-j**2-l**2+k-j-l)\n",
"\n",
" return (-S(1))**(m+p)*temp\n",
"\n",
"def alpha(l,m,j,p,k):\n",
" return sqrt((2*l+1)*(2*j+1)*(2*k+1)/(4*pi))*wigner_3j(j, l, k, 0, 0, 0)*wigner_3j(j, l, k, p, m, -m-p)\n",
"\n",
"\n",
"print dot_rota_grad_SH(2,6,2,2,0,0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"it is now incorporated in SymPy"
]
},
{
"cell_type": "code",
"execution_count": 167,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from sympy import Ynm, init_printing, Expr, var, sympify, Dummy, S, Sum, sqrt, pi\n",
"from sympy.physics.wigner import wigner_3j\n",
"# init_printing(use_latex='mathjax')"
]
},
{
"cell_type": "code",
"execution_count": 168,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"class Wigner3j(Expr):\n",
" def doit(self, **hints):\n",
" num = True\n",
" for i in range(6):\n",
" if not self.args[i].is_number:\n",
" num = False\n",
" if num:\n",
" return wigner_3j(*self.args)\n",
" else:\n",
" return self\n",
"\n",
"def alpha(l,m,j,p,k):\n",
" return sqrt((2*l+1)*(2*j+1)*(2*k+1)/(4*pi)) * Wigner3j(j, l, k, S(0), S(0), S(0))*Wigner3j(j, l, k, p, m, -m-p)\n",
"\n",
"def dot_rota_grad_SH(j, p, l, m, theta, phi):\n",
" j = sympify(j)\n",
" p = sympify(p)\n",
" l = sympify(l)\n",
" m = sympify(m)\n",
" theta = sympify(theta)\n",
" phi = sympify(phi)\n",
" k = Dummy(\"k\")\n",
" return (-S(1))**(m+p) * Sum(Ynm(k, m+p, theta, phi)*alpha(l,m,j,p,k)/2 *(k**2-j**2-l**2+k-j-l), (k, abs(l-j), l+j))"
]
},
{
"cell_type": "code",
"execution_count": 169,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$- \\sum_{k=0}^{4} \\frac{\\sqrt{50 k + 25}}{4 \\sqrt{\\pi}} \\left(k^{2} + k - 12\\right) Y_{k}^{1}\\left(\\theta,\\phi\\right) Wigner3j(2, 2, _k, 0, 0, 0) Wigner3j(2, 2, _k, 0, 1, -1)$$"
],
"text/plain": [
" 4 \n",
" _____ \n",
" ╲ \n",
" ╲ ___________ ⎛ 2 ⎞ \n",
" ╲ ╲╱ 50⋅k + 25 ⋅⎝k + k - 12⎠⋅Ynm(k, 1, θ, φ)⋅Wigner3j(2, 2, _k, 0, 0, 0)\n",
" ╲ ───────────────────────────────────────────────────────────────────────\n",
"- ╱ ___ \n",
" ╱ 4⋅╲╱ π \n",
" ╱ \n",
" ╱ \n",
" ‾‾‾‾‾ \n",
" k = 0 \n",
"\n",
" \n",
" \n",
" \n",
" \n",
"⋅Wigner3j(2, 2, _k, 0, 1, -1)\n",
"─────────────────────────────\n",
" \n",
" \n",
" \n",
" \n",
" \n",
" "
]
},
"execution_count": 169,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(\"j p l m theta phi\")\n",
"#dot_rota_grad_SH(1, 5, 1, 1, 1, 2)\n",
"dot_rota_grad_SH(2,0,2,1,theta,phi)"
]
},
{
"cell_type": "code",
"execution_count": 170,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"-3*sqrt(5)*Ynm(2, 1, theta, phi)/(14*sqrt(pi)) + 2*sqrt(30)*Ynm(4, 1, theta, phi)/(7*sqrt(pi))\n"
]
}
],
"source": [
"print _.doit()"
]
},
{
"cell_type": "code",
"execution_count": 171,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$- \\sum_{k=0}^{4} \\left(\\frac{\\sqrt{50 k + 25}}{8 \\pi} k^{2} \\sqrt{\\frac{2 k \\left(k - 1\\right)!}{\\left(k + 1\\right)!} + \\frac{\\left(k - 1\\right)!}{\\left(k + 1\\right)!}} e^{i \\phi} P_{k}^{\\left(1\\right)}\\left(\\cos{\\left (\\theta \\right )}\\right) Wigner3j(2, 2, _k, 0, 0, 0) Wigner3j(2, 2, _k, 0, 1, -1) + \\frac{\\sqrt{50 k + 25}}{8 \\pi} k \\sqrt{\\frac{2 k \\left(k - 1\\right)!}{\\left(k + 1\\right)!} + \\frac{\\left(k - 1\\right)!}{\\left(k + 1\\right)!}} e^{i \\phi} P_{k}^{\\left(1\\right)}\\left(\\cos{\\left (\\theta \\right )}\\right) Wigner3j(2, 2, _k, 0, 0, 0) Wigner3j(2, 2, _k, 0, 1, -1) - \\frac{3 e^{i \\phi}}{2 \\pi} \\sqrt{50 k + 25} \\sqrt{\\frac{2 k \\left(k - 1\\right)!}{\\left(k + 1\\right)!} + \\frac{\\left(k - 1\\right)!}{\\left(k + 1\\right)!}} P_{k}^{\\left(1\\right)}\\left(\\cos{\\left (\\theta \\right )}\\right) Wigner3j(2, 2, _k, 0, 0, 0) Wigner3j(2, 2, _k, 0, 1, -1)\\right)$$"
],
"text/plain": [
" 4 \n",
" _____ \n",
" ╲ \n",
" ╲ ⎛ _________________________ \n",
" ╲ ⎜ 2 ___________ ╱ 2⋅k⋅(k - 1)! (k - 1)! ⅈ⋅φ \n",
" ╲ ⎜k ⋅╲╱ 50⋅k + 25 ⋅ ╱ ──────────── + ──────── ⋅ℯ ⋅assoc_legendre(k, \n",
"- ╱ ⎜ ╲╱ (k + 1)! (k + 1)! \n",
" ╱ ⎜──────────────────────────────────────────────────────────────────────\n",
" ╱ ⎝ 8⋅π\n",
" ╱ \n",
" ‾‾‾‾‾ \n",
" k = 0 \n",
"\n",
" \n",
" \n",
" \n",
" \n",
" ____\n",
"1, cos(θ))⋅Wigner3j(2, 2, _k, 0, 0, 0)⋅Wigner3j(2, 2, _k, 0, 1, -1) k⋅╲╱ 50⋅\n",
" \n",
"─────────────────────────────────────────────────────────────────── + ────────\n",
" \n",
" \n",
" \n",
" \n",
"\n",
" \n",
" \n",
" \n",
" _________________________ \n",
"_______ ╱ 2⋅k⋅(k - 1)! (k - 1)! ⅈ⋅φ \n",
"k + 25 ⋅ ╱ ──────────── + ──────── ⋅ℯ ⋅assoc_legendre(k, 1, cos(θ))⋅Wigner\n",
" ╲╱ (k + 1)! (k + 1)! \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 8⋅π \n",
" \n",
" \n",
" \n",
"\n",
" \n",
" \n",
" \n",
" _____\n",
" ___________ ╱ 2⋅k⋅\n",
"3j(2, 2, _k, 0, 0, 0)⋅Wigner3j(2, 2, _k, 0, 1, -1) 3⋅╲╱ 50⋅k + 25 ⋅ ╱ ────\n",
" ╲╱ (k\n",
"────────────────────────────────────────────────── - ─────────────────────────\n",
" \n",
" \n",
" \n",
" \n",
"\n",
" \n",
" \n",
" \n",
"____________________ \n",
"(k - 1)! (k - 1)! ⅈ⋅φ \n",
"──────── + ──────── ⋅ℯ ⋅assoc_legendre(k, 1, cos(θ))⋅Wigner3j(2, 2, _k, 0, 0\n",
" + 1)! (k + 1)! \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 2⋅π \n",
" \n",
" \n",
" \n",
"\n",
" \n",
" \n",
" \n",
" ⎞\n",
" ⎟\n",
", 0)⋅Wigner3j(2, 2, _k, 0, 1, -1)⎟\n",
" ⎟\n",
"─────────────────────────────────⎟\n",
" ⎠\n",
" \n",
" \n",
" "
]
},
"execution_count": 171,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"_.expand(func=True)"
]
},
{
"cell_type": "code",
"execution_count": 173,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0 0 0 0 0\n",
"0 0 1 -1 0\n",
"0 0 1 0 0\n",
"0 0 1 1 0\n",
"1 -1 0 0 0\n",
"1 -1 1 -1 sqrt(30)*exp(-4*I*phi)*Ynm(2, 2, theta, phi)/(10*sqrt(pi))\n",
"1 -1 1 0 -sqrt(15)*exp(-2*I*phi)*Ynm(2, 1, theta, phi)/(10*sqrt(pi))\n",
"1 -1 1 1 Ynm(0, 0, theta, phi)/sqrt(pi) + sqrt(5)*Ynm(2, 0, theta, phi)/(10*sqrt(pi))\n",
"1 0 0 0 0\n",
"1 0 1 -1 -sqrt(15)*exp(-2*I*phi)*Ynm(2, 1, theta, phi)/(10*sqrt(pi))\n",
"1 0 1 0 -Ynm(0, 0, theta, phi)/sqrt(pi) + sqrt(5)*Ynm(2, 0, theta, phi)/(5*sqrt(pi))\n",
"1 0 1 1 sqrt(15)*Ynm(2, 1, theta, phi)/(10*sqrt(pi))\n",
"1 1 0 0 0\n",
"1 1 1 -1 Ynm(0, 0, theta, phi)/sqrt(pi) + sqrt(5)*Ynm(2, 0, theta, phi)/(10*sqrt(pi))\n",
"1 1 1 0 sqrt(15)*Ynm(2, 1, theta, phi)/(10*sqrt(pi))\n",
"1 1 1 1 sqrt(30)*Ynm(2, 2, theta, phi)/(10*sqrt(pi))\n"
]
}
],
"source": [
"for j in range(2):\n",
" for p in range(-j, j+1):\n",
" for l in range(2):\n",
" for m in range(-l, l+1):\n",
" print j, p, l, m, dot_rota_grad_SH(j, p, l, m, theta, phi).doit()"
]
},
{
"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 2",
"language": "python",
"name": "python2"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 2
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.10"
}
},
"nbformat": 4,
"nbformat_minor": 0
}
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