From 979d996b44a8ee01a28c6787227dac91b0e61e13 Mon Sep 17 00:00:00 2001 From: caes Date: Mon, 14 Mar 2016 01:10:25 -0400 Subject: [PATCH] working on chap 9 prob 11 --- solutions/chap9/prob11 | 99 ++ solutions/chap9/prob11.ps | 2869 +++++++++++++++++++++++++++++++++++++ 2 files changed, 2968 insertions(+) create mode 100644 solutions/chap9/prob11 create mode 100644 solutions/chap9/prob11.ps diff --git a/solutions/chap9/prob11 b/solutions/chap9/prob11 new file mode 100644 index 0000000..1079d8e --- /dev/null +++ b/solutions/chap9/prob11 @@ -0,0 +1,99 @@ +The harmonic oscillator potential is V(x) = ½kx² = ½mω²x². The hamiltonian is time independent. + +The initial state vector is + + Ψ(t=0) = A⎛❙0❭ + 2exp(ι͟π͟)❙1❭⎞ + ⎝ 2 ⎠ + +Finding the normalization constant A is simple enough, since + + 1 = A² + (2Aexp(ι͟π͟))² = A²(1 + (2exp(ι͟π͟))²). + 2 2 + + A = √⎛ ______͟1͟______ ⎞ = √⎛ ________͟1͟_______ ⎞ + ⎜ 1 + (2exp(ι͟π͟))² ⎟ ⎜ 1 + 4exp(ι͟π͟ - ι͟π͟ ) ⎟, + ⎝ 2 ⎠ ⎝ 2 2 ⎠ + +(𝐚) + A = √⎛ ____͟1͟____ ⎞ = √⅕. + ⎝ 1 + 4exp(0) ⎠ + + +The time evolution prescription for a time-independent hamiltonian is + + Ψ(t) = exp(-ι͟E͟ₙ͟t) Ψ(t=0); + ħ + + Ψ(t) = exp(-ι͟E͟ₙ͟t) √⅕⎛❙0❭ + 2exp(ι͟π͟)❙1❭⎞. + ħ ⎝ 2 ⎠ + +(𝐛) + Ψ(t) = √⅕⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞. + ⎝ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠ + +The expecation values ❬x̂❭ and ❬p̂❭ are of interest. + + ❬x̂❭ = ❬Ψ⃰(t)❙x̂❙Ψ(t)❭. + + Ψ⃰(t) = √⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞. + ⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠ + + ❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞ x̂ ⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞. + ⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠ ⎝ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠ + +The matrix representations of the position operator x̂ and momentum operator p̂ have been developed from the definition of the increment/decrement operators. The matrix elements may be ascertained by inspection. + + x̂ ≐ p̂ ≐ + ⎛ 0 √1 ⎞ ⎛ 0 -ι√1 ⎞ + ⎝ √1 0 ⎠, ⎝ ι√1 0 ⎠. + + ❬0❙x̂❙0❭ = x₀₀ = ❬1❙x̂❙1❭ = x₁₁ = 0. + + ❬0❙x̂❙1❭ = x₀₁ = ❬1❙x̂❙0❭ = x₁₀ = √1 = 1. + + + + ❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙x̂❙0❭ + ⎞ + ⎜ ⎝ ħ ⎠ ⎝ ħ ⎠ ⎟ + ⎜ ⎟ + ⎜ exp⎛ι͟E͟₀͟t⎞ 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞ ❬0❙x̂❙1❭ + ⎟ + ⎜ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎟ + ⎜ ⎟ + ⎜ exp⎛-ι͟E͟₀͟t⎞ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞ ❬1❙x̂❙0❭ + ⎟ + ⎜ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎟ + ⎜ ⎟ + ⎜ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙x̂❙1❭⎞⎟ + ⎝ ⎝ ⎝ ħ 2⎠⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠⎠. + +Substituting the matrix elements: + + ❬x̂❭ = ⅖⎛exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + ⎞ = ⅖⎛exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + ⎞ + ⎜ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎟ ⎜ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎟ + ⎜ ⎟ ⎜ ⎟ + ⎜ exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞⎟ ⎜ exp⎛ι͟⎛⎛E͟₁͟−͟E͟₀͟⎞ - _͟π͟⎞⎞⎟ + ⎝ ⎝ ħ ⎝ ħ 2⎠⎠⎠ ⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠⎠; + + ❬x̂❭ = ⅖⎛ exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ ⎞ + ⎝ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎝ ħ ⎝ ħ 2⎠⎠ ⎠; + + ❬x̂❭ = ⅖⎛ exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛E͟₁͟−͟E͟₀͟⎞t - _͟π͟⎞⎞ ⎞ + ⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎠. + +For the Harmonic Oscillator, Eₙ = ħω(n + ½). + + Eₙ−Eₙ′ = ħω(n - n′): + E₀−E₁ = ħω(0 - 1) = -ħω; + E₁−E₀ = ħω. + + ❬x̂❭ = ⅖⎛ exp⎛ι⎛⎛−͟ħ͟ω͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞ ⎞ + ⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎠. + +(𝐜,x̂) + ❬x̂❭ = ⅖⎛ ι(-ωt + _͟π͟)⎞ + exp⎛ι⎛ωt - _͟π͟⎞⎞ ⎞ + ⎝ e 2 ⎠ ⎝ ⎝ 2⎠⎠ ⎠. + +The expectation value progresses with the time parameter t. ω is a characteristic parameter of the system. 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2074 l +1263 1287 1577 752 x +1891 219 2395 219 x +2922 219 3225 769 x +3528 1319 3528 2074 x +3528 2788 l +3528 3576 3214 4109 x +2900 4644 2395 4644 x +4824 fwd_x +end_ol + } def +/xAA { start_ol +3421 1544 m +1294 1544 l +862 327 l +1494 327 l +1710 327 1710 167 x +1710 0 1494 0 x +288 0 l +72 0 72 167 x +72 327 288 327 x +535 327 l +1966 4208 l +1006 4208 l +790 4208 790 4376 x +790 4536 1006 4536 x +2637 4536 l +4212 327 l +4507 327 l +4722 327 4722 167 x +4722 0 4507 0 x +3260 0 l +3045 0 3045 167 x +3045 327 3260 327 x +3867 327 l +3421 1544 l +3300 1872 m +2405 4208 l +2286 4208 l +1423 1872 l +3300 1872 l +4824 fwd_x +end_ol + } def +/yAA { start_ol +4119 5482 m +3063 4019 3063 2134 x +3063 -1598 l +2736 -1598 l +2736 1998 l +2736 4091 3820 5641 x +3899 5738 3985 5738 x +4041 5738 4091 5690 x +4143 5641 4143 5577 x +4143 5514 4119 5482 x +4824 fwd_x +end_ol + } def +/zAA { start_ol +3008 5592 m +3008 -956 l +1803 -956 l +1803 5592 l +3008 5592 l +4824 fwd_x +end_ol + } def +/ABA { start_ol +1264 -507 m +2720 2407 l +1267 5326 l +2095 5326 l +3547 2407 l +2095 -507 l +1264 -507 l +4824 fwd_x +end_ol + } def +/CBA { start_ol +2559 2088 m +2559 500 l +2559 288 2399 288 x +2232 288 2232 500 x +2232 2088 l +855 2088 l +648 2088 648 2255 x +648 2415 855 2415 x +2232 2415 l +2232 4002 l +2232 4215 2399 4215 x +2559 4215 2559 4002 x +2559 2415 l +3992 2415 l +4215 2415 4215 2255 x +4215 2088 3992 2088 x +2559 2088 l +4824 fwd_x +end_ol + } def +/DBA { start_ol +864 3781 m +864 4143 1290 4557 x +1716 4971 2319 4971 x +2887 4971 3318 4552 x +3750 4134 3750 3587 x +3750 3225 3543 2907 x +3337 2589 2630 1904 x +1019 351 l +1019 327 l +3456 327 l +3456 615 l +3456 831 3623 831 x +3783 831 3783 615 x +3783 0 l +648 0 l +648 479 l +2525 2268 l +3082 2823 3253 3073 x +3424 3323 3424 3596 x +3424 4015 3091 4329 x +2759 4644 2319 4644 x +1926 4644 1604 4402 x +1283 4160 1191 3796 x +1141 3636 1019 3636 x +961 3636 912 3679 x +864 3724 864 3781 x +4824 fwd_x +end_ol + } def +/EBA { start_ol +3999 456 m +3999 385 3880 250 x +3762 117 3503 -13 x +3245 -144 2952 -144 x +2556 -144 2322 137 x +2088 419 2088 871 x +2088 2984 l +1639 2984 l +1440 2984 1440 3147 x +1440 3312 1643 3312 x +2415 3312 l +2415 876 l +2415 183 2954 183 x +3160 183 3322 249 x +3484 317 3562 399 x +3642 481 3718 547 x +3793 615 3840 615 x +3864 615 3900 603 x +3936 591 3967 552 x +3999 513 3999 456 x +4824 fwd_x +end_ol + } def +/FBA { start_ol +2418 -536 m +2418 -936 l +-2376 -936 l +-2376 -536 l +2418 -536 l +0 fwd_x +end_ol + } def +/GBA { start_ol +927 3312 m +3864 3312 l +4071 3312 4071 3152 x +4071 2984 3880 2984 x +3639 2984 l +3639 327 l +3880 327 l +4071 327 4071 167 x +4071 0 3875 0 x +3003 0 l +2808 0 2808 167 x +2808 327 2997 327 x +3312 327 l +3312 2984 l +1479 2984 l +1479 327 l +1794 327 l +1983 327 1983 167 x +1983 0 1787 0 x +923 0 l +720 0 720 167 x +720 327 918 327 x +1152 327 l +1152 2984 l +911 2984 l +720 2984 720 3152 x +720 3312 927 3312 x +4824 fwd_x +end_ol + } def +/HBA { start_ol +2559 4927 m +2559 327 l +3645 327 l +3855 327 3855 167 x +3855 0 3645 0 x +1154 0 l +936 0 936 167 x +936 327 1153 327 x +2232 327 l +2232 4495 l +1381 3592 l +1320 3528 1222 3528 x +1163 3528 1121 3579 x +1080 3632 1080 3712 x +1080 3784 1164 3872 x +2170 4927 l +2559 4927 l +4824 fwd_x +end_ol + } def +/IBA { start_ol +704 5511 m +680 5542 680 5607 x +680 5671 731 5718 x +783 5766 838 5766 x +877 5766 916 5743 x +956 5719 979 5686 x +1003 5663 l +2088 4128 2088 2027 x +2088 -1569 l +1760 -1569 l +1760 2162 l +1760 4048 704 5511 x +4824 fwd_x +end_ol + } def +/JBA { start_ol +4119 -687 m +4143 -718 4143 -783 x +4143 -847 4091 -894 x +4041 -942 3985 -942 x +3946 -942 3907 -919 x +3867 -895 3844 -862 x +3820 -839 l +2736 695 2736 2796 x +2736 6393 l +3063 6393 l +3063 2661 l +3063 775 4119 -687 x +4824 fwd_x +end_ol + } def +/KBA { start_ol +704 -658 m +1760 804 1760 2689 x +1760 6422 l +2088 6422 l +2088 2826 l +2088 732 1003 -817 x +924 -914 838 -914 x +783 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3420 2155 3420 x +3006 3420 3528 2650 x +2179 3092 m +1617 3092 1224 2704 x +831 2316 831 1751 x +831 1186 1224 793 x +1617 399 2179 399 x +2741 399 3134 788 x +3528 1179 3528 1735 x +3528 2315 3138 2703 x +2749 3092 2179 3092 x +4824 fwd_x +end_ol + } def +/NBA { start_ol +3783 3312 m +3783 3024 l +1383 327 l +3600 327 l +3600 765 l +3600 975 3767 975 x +3927 975 3927 762 x +3927 0 l +1008 0 l +1008 288 l +3343 2984 l +1335 2984 l +1335 2550 l +1335 2336 1175 2336 x +1008 2336 1008 2550 x +1008 3312 l +3783 3312 l +4824 fwd_x +end_ol + } def +/OBA { start_ol +3528 0 m +3528 525 l +2855 -144 2071 -144 x +1591 -144 1299 142 x +1008 430 1008 901 x +1008 2984 l +624 2984 l +432 2984 432 3152 x +432 3312 632 3312 x +1335 3312 l +1335 901 l +1335 598 1546 390 x +1759 183 2063 183 x +2864 183 3528 914 x +3528 2984 l +2953 2984 l +2736 2984 2736 3152 x +2736 3312 2955 3312 x +3855 3312 l +3855 327 l +4127 327 l +4343 327 4343 154 x +4343 0 4127 0 x +3528 0 l +4824 fwd_x +end_ol + } def +/PBA { start_ol +1654 1165 m +2718 1165 l +1559 -951 l +1455 -1152 1311 -1152 x +1216 -1152 1147 -1083 x +1080 -1015 1080 -920 x +1080 -896 1095 -832 x +1654 1165 l +4824 fwd_x +end_ol + } def +/QBA { start_ol +1014 2487 m +1454 786 l +3053 5616 l +4619 5616 l +4794 5616 4794 5472 x +4794 5328 4619 5328 x +3285 5328 l +1431 0 l +759 2160 l +255 2160 l +79 2160 79 2327 x +79 2487 255 2487 x +1014 2487 l +4824 fwd_x +end_ol + } def +/RBA { start_ol +4794 -608 m +4794 -1008 l +0 -1008 l +0 -608 l +4794 -608 l +4824 fwd_x +end_ol + } def +/SBA { start_ol +2736 6393 m +3063 6393 l +3063 -1598 l +2736 -1598 l +2736 6393 l +4824 fwd_x +end_ol + } def +/TBA { start_ol +2055 -1598 m +1728 -1598 l +1728 6393 l +2055 6393 l +2055 -1598 l +4824 fwd_x +end_ol + } def +/UBA { start_ol +3024 1328 m +864 1328 l +864 1705 l +2682 4863 l +3351 4863 l +3351 1656 l +3592 1656 l +3783 1656 3783 1488 x +3783 1328 3592 1328 x +3351 1328 l +3351 327 l +3592 327 l +3783 327 3783 167 x +3783 0 3578 0 x +2436 0 l 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x +1183 3367 1183 3244 x +2376 817 m +2376 1969 l +1382 1715 1382 1008 x +1382 769 1496 622 x +1611 475 1797 475 x +2074 475 2376 817 x +4032 fwd_x +end_ol + } def +/XBA { start_ol +3933 1626 m +3714 1626 3478 1518 x +3243 1410 3207 1410 x +3172 1410 3141 1481 x +3110 1553 3110 1602 x +3110 2850 l +4393 2850 l +4518 2850 4518 2706 x +4518 2562 4397 2562 x +3438 2562 l +3438 1770 l +3487 1786 3565 1818 x +3643 1850 3673 1861 x +3702 1874 3755 1890 x +3809 1906 3854 1910 x +3899 1914 3965 1914 x +4324 1914 4577 1636 x +4830 1359 4830 970 x +4830 542 4545 256 x +4259 -29 3834 -29 x +3579 -29 3345 66 x +3112 162 2997 265 x +2881 369 2881 425 x +2881 464 2909 527 x +2937 590 2968 590 x +3017 590 3116 507 x +3216 424 3402 340 x +3588 258 3825 258 x +4163 258 4357 460 x +4550 661 4550 978 x +4550 1287 4389 1456 x +4230 1626 3933 1626 x +3875 3812 m +1071 504 l +991 407 927 407 x +862 407 814 455 x +767 504 767 567 x +767 623 839 703 x +3636 4011 l +3723 4107 3780 4107 x +3844 4107 3891 4059 x 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fwd_x +end_ol + } def +/dBA { start_ol +1686 5375 m +1686 3368 l +2121 3816 2687 3816 x +3354 3816 3776 3290 x +4198 2765 4198 1923 x +4198 1024 3690 457 x +3182 -108 2376 -108 x +1823 -108 1302 334 x +671 -100 l +576 -100 l +576 4722 l +576 4961 497 5044 x +418 5128 144 5184 x +144 5375 l +1686 5375 l +1686 2905 m +1686 772 l +1686 480 1841 306 x +1998 131 2252 131 x +3024 131 3024 1778 x +3024 3384 2261 3384 x +1875 3384 1686 2905 x +4464 fwd_x +end_ol + } def +/eBA { start_ol +3263 148 m +3263 -308 l +3263 -852 3022 -1183 x +2783 -1515 2399 -1515 x +2016 -1515 1775 -1183 x +1535 -852 1535 -308 x +1535 148 l +1535 685 1771 988 x +2008 1292 2399 1292 x +2791 1292 3027 988 x +3263 685 3263 148 x +2595 981 m +2512 1004 2399 1004 x +2287 1004 2207 981 x +2127 957 2027 878 x +1927 799 1875 607 x +1823 415 1823 124 x +1823 -276 l +1823 -1227 2399 -1227 x +2975 -1227 2975 -276 x +2975 124 l +2975 415 2923 603 x +2872 792 2775 874 x +2679 957 2595 981 x +4824 fwd_x +end_ol + } def +/fBA { start_ol +2543 1364 m +2543 -1155 l +3105 -1155 l +3263 -1155 3263 -1299 x +3263 -1443 3105 -1443 x +1688 -1443 l +1521 -1443 1521 -1299 x +1521 -1147 1768 -1147 x +2080 -1147 2255 -1155 x +2255 973 l +1741 804 l +1678 788 1670 788 x +1622 788 1578 832 x +1535 876 1535 932 x +1535 1020 1662 1076 x +1789 1132 2543 1364 x +4824 fwd_x +end_ol + } def +/gBA { start_ol +3547 -507 m +2715 -507 l +1264 2407 l +2715 5326 l +3547 5326 l +2088 2407 l +3547 -507 l +4824 fwd_x +end_ol + } def +/hBA { start_ol +-2412 5078 m +-1356 4215 l +-1253 4127 -1253 4047 x +-1253 3983 -1300 3935 x +-1348 3888 -1413 3888 x +-1469 3888 -1557 3952 x +-2412 4655 l +-3267 3952 l +-3354 3888 -3411 3888 x +-3475 3888 -3522 3935 x +-3570 3983 -3570 4047 x +-3570 4111 -3467 4207 x +-2412 5078 l +0 fwd_x +end_ol + } def +/iBA { start_ol +-1751 6039 m +-1424 6039 l +-1447 5452 l +-896 5652 l +-801 5341 l +-1360 5189 l +-1000 4727 l +-1264 4536 l +-1584 5023 l +-1911 4536 l +-2175 4727 l +-1815 5189 l +-2374 5341 l 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3152 x +1758 2984 1542 2984 x +1127 2984 l +2437 336 l +3723 2984 l +3292 2984 l +3076 2984 3076 3152 x +3076 3312 3292 3312 x +4171 3312 l +4387 3312 4387 3152 x +4387 2984 4171 2984 x +4060 2984 l +2013 -1184 l +2533 -1184 l +2749 -1184 2749 -1344 x +2749 -1512 2533 -1512 x +671 -1512 l +455 -1512 455 -1344 x +455 -1184 671 -1184 x +1686 -1184 l +2253 -24 l +4824 fwd_x +end_ol + } def +/nBA { start_ol +2415 4110 m +2583 4110 2698 3990 x +2814 3871 2814 3703 x +2814 3543 2694 3427 x +2575 3312 2415 3312 x +2255 3312 2135 3431 x +2016 3551 2016 3711 x +2016 3871 2135 3990 x +2255 4110 2415 4110 x +4171 2088 m +623 2088 l +407 2088 407 2255 x +407 2415 623 2415 x +4171 2415 l +4387 2415 4387 2255 x +4387 2088 4171 2088 x +4171 936 m +623 936 l +407 936 407 1103 x +407 1263 623 1263 x +4171 1263 l +4387 1263 4387 1103 x +4387 936 4171 936 x +4824 fwd_x +end_ol + } def +/oBA { start_ol +3744 1220 m +3744 1561 3533 1771 x +3323 1982 3011 2050 x +2701 2120 2335 2205 x +1971 2290 1661 2383 x 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+end_ol + } def +/vBA { start_ol +3193 915 m +3337 774 l +2999 303 2666 97 x +2334 -108 1900 -108 x +1167 -108 691 434 x +216 976 216 1810 x +216 2656 741 3211 x +1267 3767 2067 3767 x +2559 3767 2891 3526 x +3223 3287 3223 2928 x +3223 2707 3089 2577 x +2956 2448 2719 2448 x +2311 2448 2304 2995 x +2304 3315 2265 3384 x +2189 3528 1983 3528 x +1686 3528 1514 3178 x +1343 2829 1343 2225 x +1343 1425 1635 950 x +1928 475 2422 475 x +2651 475 2818 573 x +2985 672 3193 915 x +3528 fwd_x +end_ol + } def +end end +%%EndPrologue +%%EndPrologue +%%Page: 1 1 +paps_bop +(+ 36000 807000AAABAACAA+ 64800 807000AAAEAAFAA+ 93600 807000GAAHAA+ 115200 807000IAAGAAJAAIAAKAAJAALAAMAA+ 180000 807000NAAIAAGAAOAA)paps_exec +(* 275900PAAFAABAAQAAGAAGAA)paps_exec +(* 515800RAAEAASAATAA+ 551800 807000GAA)paps_exec +36 797.000000 moveto 559 797.000000 lineto 0 setlinewidth stroke +(+ 36000 782936UAAVAAWAA+ 55296 782936VAAYAAZAAaAAbAAcAAdAAeAA+ 98712 782936bAAfAAeAAdAAgAAgAAYAAhAAbAAZAA+ 151776 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