- not so Frequently Asked Questions - update 2004/11/16
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種々ã®è»Œé“角é‹å‹•é‡ã¨ç£æ°—é‡åæ•°l=1,m=-1,+1çƒé¢èª¿å’Œé–¢æ•°Y[lm]ã®l,mã¯ï¼Œé‡å力å¦ã®è»Œé“角é‹å‹•é‡ã¨ç£æ°—é‡åæ•°ã§ã™ï¼Ž ã“ã“ã§ã¯ï¼Œå°ã•ã„lã«å¯¾ã™ã‚‹ã‚°ãƒ©ãƒ•ã‚’å¹¾ã¤ã‹è¡¨ç¤ºã—ã¦ã¿ã¾ã™ï¼Žã‚„ã‚Šæ–¹ã¯ï¼Œå‰é … ã¨åŒæ§˜ï¼Œåª’介変数を用ã„ã¦x,y,zを表ã™æ–¹æ³•ã§ã™ï¼Ž mãŒå¥‡æ•°ã®å ´åˆã¯çƒé¢èª¿å’Œé–¢æ•°ã¯exp(-imφ)ã®é …ã®ç‚ºã«è™šéƒ¨ã‚’æŒã¡ã¾ã™ï¼Ž m=-1,1ã®ã®ã“ã®é–¢æ•°ã¯æ¬¡ã®ã‚ˆã†ã«ãªã‚Šã¾ã™ï¼Ž Y[1,-1](t,p) = sqrt(3/8pi) sin(t) exp(-i p) Y[1, 1](t,p) = -sqrt(3/8pi) sin(t) exp( i p) |Y|^2ã¯Yã¨ãã®è¤‡ç´ 共役をã‹ã‘ã‚Œã°ã‚ˆã,上ã®2ã¤ã®å¼ã¯ã„ã¥ã‚Œã‚‚åŒã˜Y (θ)=3/8Ï€ sin^2θ ã¨ãªã‚Šã¾ã™ï¼ŽãªãŠï¼Œçƒé¢èª¿å’Œé–¢æ•°ã®è¤‡ç´ 共役ã¯ç°¡å˜ã«æ±‚ ã‚ã‚‹ã“ã¨ãŒã§ã, (Y[l,m])^* = (-1)^m Y[l,-m] ã§ä¸Žãˆã‚‰ã‚Œã¾ã™ï¼Ž ã¤ã¾ã‚Šï¼Œmã®ç¬¦å·ã‚’変ãˆï¼Œã•ã‚‰ã«mãŒå¥‡æ•°ã®å ´åˆã¯å…¨ä½“ã®ç¬¦å·ã‚’変ãˆã¾ã™ï¼Ž gnuplot> set parametric dummy variable is t for curves, u/v for surfaces gnuplot> set angle degree gnuplot> set urange [0:360] gnuplot> set vrange [-90:90] gnuplot> set isosample 36,18 gnuplot> set ticslevel 0 gnuplot> set size 0.65,1.0 gnuplot> a=3.0/(8*pi) gnuplot> fx(u,v)=cos(u)*cos(v) gnuplot> fy(u,v)=sin(u)*cos(v) gnuplot> fz(v)=sin(v) gnuplot> g(v)=cos(v)*cos(v) gnuplot> splot a*g(v)*fx(u,v),a*g(v)*fy(u,v),a*g(v)*fz(v) ![]() å·¦ãŒY[11]を自乗ã—ãŸã‚°ãƒ©ãƒ•ã§ï¼Œz軸を主軸ã«ã—ãŸãƒˆãƒ¼ãƒ©ã‚¹ã¨ãªã‚Šã¾ã™ï¼Žä½†ã— 内å´ã«ç©´ã¯ã‚ã‚Šã¾ã›ã‚“.関数ã¯sin(θ)ã‚’å«ã¿ã¾ã™ãŒï¼Œè§’度θをv㧠表ã™ã¨cos(v)ã«ãªã‚‹ã®ã§ï¼Œ g(v)=cos(v)*cos(v)ã¨é–¢æ•°å®šç¾©ã—㦠ã„ã¾ã™ï¼Ž l=2,m=-2,-1,0,+1,+2l=2ã®ã¨ãã¯ï¼Œç£æ°—é‡åæ•°m㌠-2〜2 ã¾ã§ã®5ã¤ã®é–¢æ•°ãŒã‚ã‚Šã¾ã™ï¼Žãã‚Œãžã‚Œã® 関数ã¯ä»¥ä¸‹ã§ä¸Žãˆã‚‰ã‚Œã¾ã™ï¼Ž Y[2,-2](t,p) = sqrt(15/32pi) sin(t)sin(t) exp(-2i p) Y[2,-1](t,p) = sqrt(15/8pi) sin(t)cos(t) exp(-i p) Y[2, 0](t,p) = sqrt(5/16pi) (3cos(t)cos(t)-1) Y[2, 1](t,p) = -sqrt(15/8pi) sin(t)cos(t) exp( i p) Y[2, 2](t,p) = sqrt(15/32pi) sin(t)sin(t) exp( 2i p) 自乗ã«ã—ã¦è¡¨ç¤ºã™ã‚‹ã¨|m|ãŒåŒã˜ã‚‚ã®ã¯åŒã˜ã‚°ãƒ©ãƒ•ã«ãªã‚Šã¾ã™ã®ã§ï¼Œm=0,1,2ã® 3ã¤ã®å ´åˆã«ã¤ã„ã¦è¡¨ç¤ºã—ã¾ã™ï¼Ž m=0 gnuplot> a= 5.0/(16*pi) gnuplot> g(v)= (3*sin(v)*sin(v)-1)**2 gnuplot> splot a*g(v)*fx(u,v),a*g(v)*fy(u,v),a*g(v)*fz(v) ![]() m=1 gnuplot> a=15.0/( 8*pi) gnuplot> g(v)= (sin(v)*cos(v))**2 gnuplot> splot a*g(v)*fx(u,v),a*g(v)*fy(u,v),a*g(v)*fz(v) ![]() m=2 gnuplot> a=15.0/(32*pi) gnuplot> g(v)= (cos(v)*cos(v))**2 gnuplot> splot a*g(v)*fx(u,v),a*g(v)*fy(u,v),a*g(v)*fz(v) ![]() |