From 9cdb4a023a6fc3666d1396f8976a3d3eb02c330a Mon Sep 17 00:00:00 2001 From: othocaes Date: Wed, 23 Mar 2016 19:43:48 -0400 Subject: [PATCH] new overview and HW, prob 8 --- HW | 2 +- lecture_notes/3-23/overview | 35 + solutions/chap7/prob7 | 40 +- solutions/chap7/prob7.ps | 1337 ++++++++++++++----- solutions/chap7/prob8 | 40 + solutions/chap7/prob8.ps | 2452 +++++++++++++++++++++++++++++++++++ 6 files changed, 3553 insertions(+), 353 deletions(-) create mode 100644 lecture_notes/3-23/overview create mode 100644 solutions/chap7/prob8 create mode 100644 solutions/chap7/prob8.ps diff --git a/HW b/HW index b71b1e9..31f7e4a 100644 --- a/HW +++ b/HW @@ -7,4 +7,4 @@ Chap 5: 2, 6, 8, 12, in-class assignment Chap 9: 7, 11, 12, 13, 14 Chap 7: 7, 8, one from last class, 11 - due friday 3-18 + due friday 3-18, the class one is: show L̂𝓏 and p² commute diff --git a/lecture_notes/3-23/overview b/lecture_notes/3-23/overview new file mode 100644 index 0000000..9781edc --- /dev/null +++ b/lecture_notes/3-23/overview @@ -0,0 +1,35 @@ +Two dimensional harmonic oscillator +─────────────────────────────────── + + This is an oscillator with potential V(x,y) = μ/2 ω (x² + y²) + + The hamiltonian here leaves us with a 3-dimensional differential equation + + Ĥ = ιp²/2μ + μ/2 ω² (x² + y²) = 1/2μ (p²𝓍 + p²𝓎 + p²𝓏) + μ/2 ω²(x² + y²) + + (pic) This is then split into x,y, and z parts. + + (pic) Solved Ψ(z) + + + Put together solutions of Ψ(x,y) and Ψ(z). + + (pic) To find position space representation of Ψ(x,y), recall the Hermitian Polynomials solution + + !!! STUDY THIS !!! + + Developed the harmonic oscillator in polar coordinates + + Ĥ𝓍𝓎 = -ħ²\2μ ∇² + μ/2 ω² r² + + (pic) Can be solved using separation of variables. + + Ψ(r,θ) = R(r) Θ(θ) + + The problem is invariant under rotations about z, i.e. L̂𝓏 commutes with Ĥ, so the solutions must be eigenstates of both Ĥ and L̂𝓏. + + Don't use L̂𝓏 Θ = ±ιmħ Θ + + Use L̂𝓏² Θ = -m²ħ² Θ ⇒ Θ(θ) = exp(±imθ) + + (pic) further developed hamiltonian using this information \ No newline at end of file diff --git a/solutions/chap7/prob7 b/solutions/chap7/prob7 index 2afa48b..e84ac89 100644 --- a/solutions/chap7/prob7 +++ b/solutions/chap7/prob7 @@ -1,6 +1,6 @@ Angular momentum system is prepared in the state - ❙Ψ❭ = 1͟ ❙11❭ - 2͟ ❙10❭ + ι͟2͟ ❙22❭ + ι1͟ ❙20❭ + ❙Ψ❭ = 1͟ ❙11❭ - 2͟ ❙10❭ + ι͟2͟ ❙22❭ + ι͟1͟ ❙20❭ √10 √10 √10 √10 @@ -18,7 +18,7 @@ For lm = 22 & 20, 𝐋̂² = 6ħ². Since the same 𝐋̂² is measures for two states, the sum of the probabilities of measuring those states is the probability of measuring that squared angular momentum. -𝓟(𝐋̂²=2ħ²) = │❬11❙Ψ❭│² + │❬10❙Ψ❭│². + 𝓟(𝐋̂²=2ħ²) = │❬11❙Ψ❭│² + │❬10❙Ψ❭│². ❬11❙Ψ❭ = ❬11❙⎛ 1͟ ❙11❭ - 2͟ ❙10❭ + ι͟2͟ ❙22❭ + ι1͟ ❙20❭ ⎞ ⎝ √10 √10 √10 √10 ⎠. @@ -41,29 +41,29 @@ No need to calculate the other set: since the vector is normalized, the probabil 𝓟(𝐋̂²=6ħ²) = ½. -For 𝐋̂𝓏 the eigenvalue equation is +For L̂𝓏 the eigenvalue equation is - 𝐋̂𝓏❙lm❭ = mħ❙lm❭, + L̂𝓏❙lm❭ = mħ❙lm❭, so the expected measurements are, -for lm = 11 & 21, 𝐋̂𝓏 = ħ. +for lm = 11 & 21, L̂𝓏 = ħ. -for lm = 10 & 20, 𝐋̂𝓏 = 0. +for lm = 10 & 20, L̂𝓏 = 0. In this case, the probabilities will be - 𝓟(𝐋̂𝓏=ħ) = │❬11❙Ψ❭│² + │❬21❙Ψ❭│² and + 𝓟(L̂𝓏=ħ) = │❬11❙Ψ❭│² + │❬21❙Ψ❭│² and - 𝓟(𝐋̂𝓏=0) = │❬10❙Ψ❭│² + │❬20❙Ψ❭│². + 𝓟(L̂𝓏=0) = │❬10❙Ψ❭│² + │❬20❙Ψ❭│². The method has already been demonstrated, so taking the probability components and summing, (𝐛) - 𝓟(𝐋̂𝓏=ħ) = ²/₁₀. + 𝓟(L̂𝓏=ħ) = ²/₁₀. - 𝓟(𝐋̂𝓏=0) = ⁸/₁₀. + 𝓟(L̂𝓏=0) = ⁸/₁₀. Histogram of probabilities: @@ -72,17 +72,17 @@ Histogram of probabilities: ╭─────────────────┬────────────────╮ 1 │ │ │ 1 │ │ │ -.8 │ │ ▧ │ .8 - │ │ ▧ │ - │ │ ▧ │ -.5 │ ▧        ▧    │           ▧    │ .5 -   │   ▧        ▧    │           ▧    │     -   │   ▧        ▧    │           ▧    │     -   │   ▧        ▧    │   ▧       ▧    │     -.1 │   ▧        ▧    │   ▧       ▧    │ .1 +.8 │ │ ▓ │ .8 + │ │ ▓ │ + │ │ ▓ │ +.5 │ ▓        ▓    │           ▓    │ .5 +   │   ▓        ▓    │           ▓    │     +   │   ▓        ▓    │           ▓    │     +   │   ▓        ▓    │   ▓       ▓    │     +.1 │   ▓        ▓    │   ▓       ▓    │ .1    ╰─────────────────┼────────────────╯ - 2ħ² 6ħ² │ 0 ħ - 𝐋̂² │ 𝐋̂𝓏 + 2ħ² 6ħ² 0 ħ + 𝐋̂² L̂𝓏 h diff --git a/solutions/chap7/prob7.ps b/solutions/chap7/prob7.ps index f126baa..9cbf4e0 100644 --- a/solutions/chap7/prob7.ps +++ b/solutions/chap7/prob7.ps @@ -178,6 +178,77 @@ end /fontdict 1 dict def papsdict begin fontdict begin /AAA { start_ol +0 8784 m +1522 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+1076,7 @@ end_ol 5400 fwd_x end_ol } def -/iAA { start_ol +/lAA { start_ol 1017 3744 m 1017 3276 l 1237 3610 1450 3748 x @@ -1023,7 +1119,7 @@ end_ol 5400 fwd_x end_ol } def -/jAA { start_ol +/mAA { start_ol 2704 3888 m 3570 3888 4165 3297 x 4761 2706 4761 1849 x @@ -1045,7 +1141,7 @@ end_ol 5400 fwd_x end_ol } def -/kAA { start_ol +/nAA { start_ol 4680 1800 m 936 1800 l 1035 1084 1530 654 x @@ -1072,7 +1168,7 @@ end_ol 5400 fwd_x end_ol } def -/lAA { start_ol +/oAA { start_ol 1665 3744 m 3588 3744 l 3825 3744 3825 3564 x @@ -1102,7 +1198,7 @@ end_ol 5400 fwd_x end_ol } def -/mAA { start_ol +/pAA { start_ol 4032 1054 m 4032 1387 3744 1567 x 3456 1747 3024 1813 x @@ -1144,7 +1240,7 @@ end_ol 5400 fwd_x end_ol } def -/nAA { start_ol +/qAA { start_ol 2538 10 m 846 3375 l 702 3375 l @@ -1175,7 +1271,7 @@ end_ol 5400 fwd_x end_ol } def -/oAA { start_ol +/rAA { start_ol 2889 3744 m 2889 369 l 4298 369 l @@ -1198,7 +1294,7 @@ end_ol 5400 fwd_x end_ol } def -/pAA { start_ol +/sAA { start_ol 1305 3744 m 1305 3004 l 1660 3483 2023 3685 x @@ -1233,7 +1329,7 @@ end_ol 5400 fwd_x end_ol } def -/qAA { start_ol +/tAA { start_ol 4545 5409 m 4545 369 l 5025 369 l @@ -1264,7 +1360,7 @@ end_ol 5400 fwd_x end_ol } def -/rAA { start_ol +/uAA { start_ol 3960 2598 m 3960 3004 3655 3261 x 3350 3519 2857 3519 x @@ -1301,7 +1397,7 @@ end_ol 5400 fwd_x end_ol } def -/tAA { start_ol +/wAA { start_ol 3388 6297 m 3388 -1076 l 2030 -1076 l @@ -1310,7 +1406,7 @@ end_ol 5400 fwd_x end_ol } def -/uAA { start_ol +/xAA { start_ol 3108 1656 m 3521 1656 3920 2098 x 4320 2541 4320 3153 x @@ -1365,7 +1461,7 @@ end_ol 5400 fwd_x end_ol } def -/vAA { start_ol +/yAA { start_ol 1424 -571 m 3063 2711 l 1428 5998 l @@ -1376,7 +1472,7 @@ end_ol 5400 fwd_x end_ol } def -/wAA { start_ol +/zAA { start_ol 4698 3024 m 702 3024 l 459 3024 459 3213 x @@ -1394,7 +1490,7 @@ end_ol 5400 fwd_x end_ol } def -/xAA { start_ol +/ABA { start_ol 2889 5535 m 2889 369 l 4095 369 l @@ -1415,7 +1511,7 @@ end_ol 5400 fwd_x end_ol } def -/yAA { start_ol +/BBA { start_ol 2736 -630 m 2736 -1080 l -2664 -1080 l @@ -1424,7 +1520,7 @@ end_ol 0 fwd_x end_ol } def -/zAA { start_ol +/CBA { start_ol 4509 2304 m 891 2304 l 648 2304 648 2493 x @@ -1435,7 +1531,7 @@ end_ol 5400 fwd_x end_ol } def -/ABA { start_ol +/DBA { start_ol 936 4223 m 936 4627 1431 5089 x 1927 5553 2628 5553 x @@ -1465,7 +1561,7 @@ end_ol 5400 fwd_x end_ol } def -/BBA { start_ol +/EBA { start_ol 4401 3150 m 4401 2250 l 4401 1188 3929 522 x @@ -1491,7 +1587,7 @@ end_ol 5400 fwd_x end_ol } def -/CBA { start_ol +/FBA { start_ol 2889 2376 m 2889 597 l 2889 360 2709 360 x @@ -1512,7 +1608,7 @@ end_ol 5400 fwd_x end_ol } def -/DBA { start_ol +/GBA { start_ol 4545 490 m 4545 415 4410 273 x 4275 131 3982 -5 x @@ -1536,7 +1632,7 @@ end_ol 5400 fwd_x end_ol } def -/EBA { start_ol +/HBA { start_ol 1143 2817 m 1638 891 l 3438 6300 l @@ -1553,7 +1649,7 @@ end_ol 5400 fwd_x end_ol } def -/FBA { start_ol +/IBA { start_ol 1521 2079 m 1521 369 l 2718 369 l @@ -1584,7 +1680,7 @@ end_ol 5400 fwd_x end_ol } def -/GBA { start_ol +/JBA { start_ol 1305 5409 m 1305 2959 l 1953 3888 2902 3888 x @@ -1615,7 +1711,7 @@ end_ol 5400 fwd_x end_ol } def -/HBA { start_ol +/KBA { start_ol 4648 4968 m 4630 4968 4428 4986 x 4225 5004 3939 5022 x @@ -1649,7 +1745,7 @@ end_ol 5400 fwd_x end_ol } def -/IBA { start_ol +/LBA { start_ol 3114 270 m 3375 270 3604 306 x 3834 342 4014 394 x @@ -1681,7 +1777,7 @@ end_ol 5976 fwd_x end_ol } def -/JBA { start_ol +/MBA { start_ol 1656 5799 m 1656 6034 1945 6311 x 2234 6588 2641 6588 x @@ -1709,7 +1805,7 @@ end_ol 5400 fwd_x end_ol } def -/KBA { start_ol +/NBA { start_ol 1224 4750 m 1314 4777 1593 4908 x 1872 5040 2146 5112 x @@ -1745,7 +1841,7 @@ end_ol 5400 fwd_x end_ol } def -/LBA { start_ol +/OBA { start_ol 2889 369 m 3805 369 l 4041 369 4041 189 x @@ -1771,7 +1867,7 @@ end_ol 5400 fwd_x end_ol } def -/MBA { start_ol +/PBA { start_ol -2727 5733 m -1539 4761 l -1422 4662 -1422 4572 x @@ -1788,7 +1884,7 @@ end_ol 0 fwd_x end_ol } def -/NBA { start_ol +/QBA { start_ol 3969 0 m 3520 0 l 2703 2324 l @@ -1818,7 +1914,7 @@ end_ol 5400 fwd_x end_ol } def -/OBA { start_ol +/RBA { start_ol 3024 0 m 2394 0 l 882 3375 l @@ -1843,7 +1939,7 @@ end_ol 5400 fwd_x end_ol } def -/PBA { start_ol +/SBA { start_ol 4833 776 m 4833 697 4680 551 x 4527 405 4280 245 x @@ -1873,7 +1969,7 @@ end_ol 5400 fwd_x end_ol } def -/QBA { start_ol +/TBA { start_ol 2961 1966 m 4617 369 l 4698 369 l @@ -1914,7 +2010,7 @@ end_ol 5400 fwd_x end_ol } def -/RBA { start_ol +/UBA { start_ol 2637 1035 m 2727 1035 l 2988 1035 3168 868 x @@ -1929,7 +2025,7 @@ end_ol 5400 fwd_x end_ol } def -/SBA { start_ol +/VBA { start_ol 4176 3004 m 4176 3744 l 5024 3744 l @@ -1963,7 +2059,7 @@ end_ol 5400 fwd_x end_ol } def -/UBA { start_ol +/XBA { start_ol 2664 2187 m 2664 2853 2926 3649 x 3189 4446 3481 4941 x @@ -1984,7 +2080,7 @@ end_ol 5400 fwd_x end_ol } def -/VBA { start_ol +/YBA { start_ol 1368 -936 m 1368 -837 1501 -571 x 1636 -306 1800 13 x @@ -2005,7 +2101,7 @@ end_ol 5400 fwd_x end_ol } def -/WBA { start_ol +/ZBA { start_ol 3960 2598 m 3960 3004 3655 3261 x 3350 3519 2857 3519 x @@ -2052,7 +2148,7 @@ end_ol 5400 fwd_x end_ol } def -/YBA { start_ol +/bBA { start_ol 1872 1314 m 3069 1314 l 1764 -1071 l @@ -2064,7 +2160,7 @@ end_ol 5400 fwd_x end_ol } def -/ZBA { start_ol +/cBA { start_ol 1521 2448 m 1521 369 l 2718 369 l @@ -2097,7 +2193,7 @@ end_ol 5400 fwd_x end_ol } def -/aBA { start_ol +/dBA { start_ol 3555 0 m 3276 432 l 2871 -144 2268 -144 x @@ -2141,7 +2237,7 @@ end_ol 5400 fwd_x end_ol } def -/bBA { start_ol +/eBA { start_ol 4419 5112 m 4392 5112 4242 5148 x 4093 5184 3841 5184 x @@ -2178,7 +2274,7 @@ end_ol 5400 fwd_x end_ol } def -/cBA { start_ol +/fBA { start_ol 4176 1368 m 4176 1742 3946 1974 x 3717 2207 3377 2282 x @@ -2221,7 +2317,7 @@ end_ol 5400 fwd_x end_ol } def -/dBA { start_ol +/gBA { start_ol 4135 5886 m 5079 5886 5729 5537 x 6381 5188 6381 4527 x @@ -2262,7 +2358,7 @@ end_ol 5904 fwd_x end_ol } def -/eBA { start_ol +/hBA { start_ol 2520 7200 m 2889 7200 l 2889 -1800 l @@ -2271,7 +2367,7 @@ end_ol 5400 fwd_x end_ol } def -/fBA { start_ol +/iBA { start_ol 3994 -571 m 3058 -571 l 1424 2711 l @@ -2282,7 +2378,7 @@ end_ol 5400 fwd_x end_ol } def -/gBA { start_ol +/jBA { start_ol 4663 6174 m 3465 4527 3465 2403 x 3465 -1800 l @@ -2296,7 +2392,7 @@ end_ol 5400 fwd_x end_ol } def -/hBA { start_ol +/kBA { start_ol 736 6174 m 711 6210 711 6282 x 711 6354 768 6408 x @@ -2312,7 +2408,7 @@ end_ol 5400 fwd_x end_ol } def -/iBA { start_ol +/lBA { start_ol 4663 -774 m 4689 -810 4689 -882 x 4689 -954 4630 -1008 x @@ -2328,7 +2424,7 @@ end_ol 5400 fwd_x end_ol } def -/jBA { start_ol +/mBA { start_ol 736 -774 m 1935 873 1935 2997 x 1935 7200 l @@ -2342,7 +2438,7 @@ end_ol 5400 fwd_x end_ol } def -/kBA { start_ol +/nBA { start_ol 2025 4671 m 2025 369 l 4536 369 l @@ -2365,7 +2461,7 @@ end_ol 5400 fwd_x end_ol } def -/lBA { start_ol +/oBA { start_ol 2142 5409 m 3294 5409 l 2988 3132 l @@ -2375,7 +2471,7 @@ end_ol 5400 fwd_x end_ol } def -/mBA { start_ol +/pBA { start_ol 2889 4671 m 2889 369 l 4087 369 l @@ -2396,7 +2492,7 @@ end_ol 5400 fwd_x end_ol } def -/nBA { start_ol +/qBA { start_ol 1521 2448 m 1521 369 l 4320 369 l @@ -2431,7 +2527,7 @@ end_ol 5400 fwd_x end_ol } def -/oBA { start_ol +/rBA { start_ol 2637 1035 m 2727 1035 l 2988 1035 3168 868 x @@ -2457,7 +2553,7 @@ end_ol 5400 fwd_x end_ol } def -/pBA { start_ol +/sBA { start_ol 2448 5445 m 3573 5445 l 2340 3294 l @@ -2469,7 +2565,7 @@ end_ol 5400 fwd_x end_ol } def -/qBA { start_ol +/tBA { start_ol 1377 4720 m 1377 4426 1746 4233 x 2116 4041 2688 4041 x @@ -2511,7 +2607,7 @@ end_ol 5400 fwd_x end_ol } def -/rBA { start_ol +/uBA { start_ol 2830 981 m 2830 678 2953 522 x 3075 365 3322 365 x @@ -2528,7 +2624,7 @@ end_ol 5400 fwd_x end_ol } def -/sBA { start_ol +/vBA { start_ol 2860 2458 m 2953 2638 3098 2721 x 3243 2804 3449 2804 x @@ -2560,7 +2656,7 @@ end_ol 5400 fwd_x end_ol } def -/tBA { start_ol +/wBA { start_ol 2844 6552 m 2844 3708 l 3461 3708 l @@ -2580,7 +2676,7 @@ end_ol 5400 fwd_x end_ol } def -/uBA { start_ol +/xBA { start_ol 4356 5697 m 1377 -567 l 1298 -729 1183 -729 x @@ -2595,7 +2691,7 @@ end_ol 5400 fwd_x end_ol } def -/vBA { start_ol +/yBA { start_ol 2871 1521 m 2871 -1323 l 3488 -1323 l @@ -2615,7 +2711,7 @@ end_ol 5400 fwd_x end_ol } def -/wBA { start_ol +/zBA { start_ol 3663 171 m 3663 -351 l 3663 -972 3395 -1350 x @@ -2643,7 +2739,7 @@ end_ol 5400 fwd_x end_ol } def -/xBA { start_ol +/ACA { start_ol 3024 6255 m 2070 4644 l 3024 4644 l @@ -2674,7 +2770,7 @@ end_ol 5400 fwd_x end_ol } def -/yBA { start_ol +/BCA { start_ol 1287 3627 m 1287 3582 1395 3388 x 1503 3195 1503 3042 x @@ -2717,7 +2813,7 @@ end_ol 4536 fwd_x end_ol } def -/zBA { start_ol +/CCA { start_ol 4353 4325 m 1179 568 l 1089 459 1017 459 x @@ -2772,7 +2868,7 @@ end_ol 5400 fwd_x end_ol } def -/ACA { start_ol +/DCA { start_ol 4473 0 m 4006 0 l 1305 4510 l @@ -2802,7 +2898,7 @@ end_ol 5400 fwd_x end_ol } def -/BCA { start_ol +/ECA { 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sinθ sinϕ ∂͟_ ) +⎩ ∂y ∂y diff --git a/solutions/chap7/prob8.ps b/solutions/chap7/prob8.ps new file mode 100644 index 0000000..5290545 --- /dev/null +++ b/solutions/chap7/prob8.ps @@ -0,0 +1,2452 @@ +%!PS-Adobe-3.0 +%%Title: Otho Ulrich: Chap 7 #8 +%%Creator: paps version 0.6.7 by Dov Grobgeld +%%Pages: (atend) +%%BoundingBox: 0 0 595 841 +%%BeginProlog +%%Orientation: Portrait +/papsdict 1 dict def +papsdict begin + +/inch {72 mul} bind def +/mm {1 inch 25.4 div mul} bind def + +% override setpagedevice if it is not defined +/setpagedevice where { + pop % get rid of its dictionary + /setpagesize { + 3 dict begin + /pageheight exch def + /pagewidth exch def + /orientation 0 def + % Exchange pagewidth and pageheight so that pagewidth is bigger + pagewidth pageheight gt { + pagewidth + /pagewidth pageheight def + /pageheight exch def + /orientation 3 def + } if + 2 dict + dup /PageSize [pagewidth pageheight] put + dup /Orientation orientation put + setpagedevice + end + } def +} +{ + /setpagesize { pop pop } def +} ifelse +/duplex { + statusdict /setduplexmode known + { statusdict begin setduplexmode end } {pop} ifelse +} def +/tumble { + statusdict /settumble known + { statusdict begin settumble end } {pop} ifelse +} def +% Turn the page around +/turnpage { + 90 rotate + 0 pageheight neg translate +} def +% User settings +/pagewidth 595 def +/pageheight 841 def +pagewidth pageheight setpagesize +/column_width 523 def +/bodyheight 755 def +/lmarg 36 def +/ytop 791 def +/do_separation_line true def +/do_landscape false def +/do_tumble true def +/do_duplex true def +% Procedures to translate position to first and second column +/lw 20 def % whatever +/setnumcolumns { + /numcolumns exch def + /firstcolumn { /xpos lmarg def /ypos ytop def} def + /nextcolumn { + do_separation_line { + xpos column_width add gutter_width 2 div add % x start + ytop lw add moveto % y start + 0 bodyheight lw add neg rlineto 0 setlinewidth stroke + } if + /xpos xpos column_width add gutter_width add def + /ypos ytop def + } def +} def + +1 setnumcolumns +/showline { + /y exch def + /s exch def + xpos y moveto + column_width 0 rlineto stroke + xpos y moveto /Helvetica findfont 20 scalefont setfont s show +} def +/paps_bop { % Beginning of page definitions + papsdict begin + gsave + do_landscape {turnpage} if + % ps2pdf gets wrong orientation without this! + /Helvetica findfont setfont 100 100 moveto ( ) show + firstcolumn + end +} def + +/paps_eop { % End of page cleanups + grestore +} def +%%BeginProlog +/papsdict 1 dict def +papsdict begin + +/conicto { + /to_y exch def + /to_x exch def + /conic_cntrl_y exch def + /conic_cntrl_x exch def + currentpoint + /p0_y exch def + /p0_x exch def + /p1_x p0_x conic_cntrl_x p0_x sub 2 3 div mul add def + /p1_y p0_y conic_cntrl_y p0_y sub 2 3 div mul add def + /p2_x p1_x to_x p0_x sub 1 3 div mul add def + /p2_y p1_y to_y p0_y sub 1 3 div mul add def + p1_x p1_y p2_x p2_y to_x to_y curveto +} bind def +/start_ol { gsave } bind def +/end_ol { closepath fill grestore } bind def +/draw_char { fontdict begin gsave 0.001000 dup scale last_x last_y translate load exec end grestore} def +/goto_xy { fontdict begin /last_y exch def /last_x exch def end } def +/goto_x { fontdict begin /last_x exch def end } def +/fwd_x { fontdict begin /last_x exch last_x add def end } def +/c /curveto load def +/x /conicto load def +/l /lineto load def +/m /moveto load def +end +/paps_exec { + 1 dict begin + /ps exch def + /len ps length def + /pos 0 def + + % Loop over all the characters of the string + { + pos len eq {exit} if + + % Get character at pos + /ch ps pos 1 getinterval def + + % check for + + (+) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /yp ps pos 8 add 8 getinterval cvi def + /pos 16 pos add def + papsdict begin xp yp goto_xy end + } { + (*) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /pos 8 pos add def + papsdict begin xp goto_x end + } { (>) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp 2 mul fwd_x end + } { (-) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp neg 2 mul fwd_x end + } { + % Must be a 3 char sym. Load and exec + /name ps pos 3 getinterval def + papsdict begin name draw_char end + /pos 3 pos add def + } ifelse + } ifelse + } ifelse + } ifelse + } loop + end +} def +/fontdict 1 dict def +papsdict begin fontdict begin +/AAA { start_ol +0 8784 m +1522 8784 l +2153 2326 l +2913 6480 l +4358 6480 l +5243 2326 l +5738 8784 l +7272 8784 l +6257 0 l +4636 0 l +3632 4593 l +2695 0 l +1085 0 l +0 8784 l +7200 fwd_x +end_ol + } def +/BAA { start_ol +6480 360 m +5883 108 5263 -18 x +4644 -144 3954 -144 x +2311 -144 1443 731 x +576 1606 576 3252 x +576 4846 1413 5771 x +2251 6696 3697 6696 x +5157 6696 5962 5823 x +6768 4950 6768 3364 x +6768 2664 l +2304 2664 l +2309 1938 2760 1580 x +3213 1224 4110 1224 x +4702 1224 5277 1396 x +5852 1569 6480 1944 x +6480 360 l +5040 3960 m +5028 4635 4695 4981 x +4363 5328 3717 5328 x +3133 5328 2786 4970 x +2440 4612 2376 3954 x +5040 3960 l +7200 fwd_x +end_ol + } def +/CAA { start_ol +4608 5598 m +4608 9144 l +6336 9144 l +6336 0 l +4608 0 l +4608 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x +5121 4125 3736 2724 x +2985 1965 2543 1512 x +7200 fwd_x +end_ol + } def +/IAA { start_ol +3240 3960 m +2304 3960 l +2304 5472 l +3240 5472 l +3891 5472 4249 5728 x +4608 5985 4608 6450 x +4608 6940 4249 7213 x +3891 7488 3240 7488 x +2745 7488 2179 7358 x +1614 7230 1008 6984 x +1008 8568 l +1613 8742 2200 8834 x +2787 8928 3327 8928 x +4713 8928 5488 8327 x +6264 7726 6264 6670 x +6264 5893 5821 5401 x +5380 4908 4560 4756 x +5496 4592 5988 4007 x +6480 3422 6480 2470 x +6480 1195 5658 525 x +4837 -144 3272 -144 x +2604 -144 1957 -32 x +1310 77 720 288 x +720 1872 l +1278 1625 1923 1496 x +2570 1368 3281 1368 x +3993 1368 4408 1690 x +4824 2014 4824 2561 x +4824 3232 4410 3595 x +3996 3960 3240 3960 x +7200 fwd_x +end_ol + } def +/JAA { start_ol +1080 1512 m +3024 1512 l +3024 7200 l +1224 6768 l +1224 8352 l +3021 8784 l +4680 8784 l +4680 1512 l +6624 1512 l +6624 0 l +1080 0 l +1080 1512 l +7200 fwd_x +end_ol + } def +/KAA { start_ol +2880 4388 m +2880 4695 3088 4903 x +3296 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start_ol +3810 4464 m +3251 4464 2957 4047 x +2664 3631 2664 2840 x +2664 2056 2957 1640 x +3251 1224 3810 1224 x +4368 1224 4667 1640 x +4968 2056 4968 2840 x +4968 3625 4667 4044 x +4368 4464 3810 4464 x +6048 8568 m +6048 6984 l +5591 7272 5171 7416 x +4752 7560 4363 7560 x +3438 7560 2954 6933 x +2471 6306 2448 5081 x +2722 5456 3148 5644 x +3576 5832 4155 5832 x +5337 5832 5980 5081 x +6624 4332 6624 2955 x +6624 1479 5897 667 x +5170 -144 3852 -144 x +2245 -144 1518 927 x +792 1999 792 4388 x +792 6661 1674 7794 x +2557 8928 4323 8928 x +4732 8928 5168 8838 x +5603 8748 6048 8568 x +7200 fwd_x +end_ol + } def +/OAA { start_ol +3668 7344 m +2995 7344 2685 6645 x +2376 5947 2376 4388 x +2376 2836 2685 2137 x +2995 1440 3668 1440 x +4348 1440 4657 2137 x +4968 2836 4968 4388 x +4968 5947 4657 6645 x +4348 7344 3668 7344 x +576 4388 m +576 6631 1357 7779 x +2139 8928 3668 8928 x +5204 8928 5986 7779 x +6768 6631 6768 4388 x +6768 2152 5986 1003 x +5204 -144 3668 -144 x +2139 -144 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5773 x +6696 4852 6696 3276 x +6696 1699 5866 777 x +5037 -144 3632 -144 x +2234 -144 1405 777 x +576 1699 576 3276 x +7200 fwd_x +end_ol + } def +/SAA { start_ol +648 3271 m +648 8784 l +2376 8784 l +2376 2838 l +2376 2183 2710 1811 x +3044 1440 3636 1440 x +4227 1440 4561 1811 x +4896 2183 4896 2838 x +4896 8784 l +6624 8784 l +6624 3271 l +6624 1481 5915 668 x +5206 -144 3636 -144 x +2072 -144 1360 668 x +648 1481 648 3271 x +7200 fwd_x +end_ol + } def +/TAA { start_ol +2304 2729 m +2304 7848 l +576 7848 l +576 9144 l +4032 9144 l +4032 2729 l +4032 1947 4273 1621 x +4516 1296 5096 1296 x +6480 1296 l +6480 0 l +4611 0 l +3378 0 2840 631 x +2304 1262 2304 2729 x +7200 fwd_x +end_ol + } def +/UAA { start_ol +1296 6552 m +4680 6552 l +4680 1296 l +6840 1296 l +6840 0 l +792 0 l +792 1296 l +2952 1296 l +2952 5256 l +1296 5256 l +1296 6552 l +2952 9792 m +4680 9792 l +4680 7776 l +2952 7776 l +2952 9792 l +7200 fwd_x +end_ol + } def +/VAA { start_ol +6264 360 m +5828 108 5328 -18 x 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l +2871 5616 m +2871 4680 l +2376 4680 l +2376 5616 l +2871 5616 l +5400 fwd_x +end_ol + } def +/jAA { start_ol +4032 1054 m +4032 1387 3744 1567 x +3456 1747 3024 1813 x +2592 1881 2160 1948 x +1728 2016 1440 2242 x +1152 2469 1152 2868 x +1152 3310 1568 3598 x +1984 3888 2620 3888 x +3362 3888 3816 3474 x +3816 3507 l +3816 3744 4038 3744 x +4185 3744 4185 3495 x +4185 2883 l +4185 2643 4038 2643 x +3847 2643 3816 2850 x +3781 3157 3466 3337 x +3152 3519 2655 3519 x +2166 3519 1843 3323 x +1521 3128 1521 2837 x +1521 2565 1827 2418 x +2133 2273 2562 2201 x +2993 2130 3426 2058 x +3861 1986 4131 1731 x +4401 1476 4401 1046 x +4401 532 3911 193 x +3422 -144 2673 -144 x +1836 -144 1305 342 x +1305 243 l +1305 0 1152 0 x +936 0 936 243 x +936 992 l +936 1235 1152 1235 x +1305 1235 1305 1037 x +1305 973 l +1305 667 1696 445 x +2089 225 2647 225 x +3241 225 3636 459 x +4032 693 4032 1054 x +5400 fwd_x +end_ol + } def +/kAA { start_ol +2884 3519 m +2722 3519 2574 3486 x +2426 3454 2296 3386 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