From 9020e8463c87376af8985cd57b7c3e1b70a5985b Mon Sep 17 00:00:00 2001 From: caes Date: Sun, 28 Feb 2016 17:15:29 -0500 Subject: [PATCH] working on chap 5 homework --- notes/fourier | 10 + notes/particles in a box | 14 +- solutions/chap5/prob2 | 100 +- solutions/chap5/prob2.ps | 2327 ++++++++++++++++++++++++++++---------- 4 files changed, 1822 insertions(+), 629 deletions(-) create mode 100644 notes/fourier diff --git a/notes/fourier b/notes/fourier new file mode 100644 index 0000000..f1c6e70 --- /dev/null +++ b/notes/fourier @@ -0,0 +1,10 @@ +general series + + ∞ +f(x) ~ ∑ cₙ φₙ(x) + ⁿ⁼⁰ + +cₙ = ❬͟f͟,φ͟ₙ͟❭͟ᵥ where v is some modulo weighting + ‖φₙ‖ᵥ² + + diff --git a/notes/particles in a box b/notes/particles in a box index dc22f79..a9bb182 100644 --- a/notes/particles in a box +++ b/notes/particles in a box @@ -47,21 +47,23 @@ Energy eigenstates obey the following properties: Bra-ket Notation Wavefunction Notation - Normalization + Normalization ⌠∞ ❬Eₙ❙Eₙ❭ = 1 ⎮ │φₙ(x)│² dx = 1 ⌡-∞ - Orthogonality + Orthogonality ⌠∞ ❬Eₙ❙Eₘ❭ = δₙₘ ⎮ φₙ⃰(x) φₘ(x) dx = δₙₘ ⌡-∞ - Completeness - ⌠∞ - ❬Eₙ❙Eₘ❭ = δₙₘ ⎮ φₙ⃰(x) φₘ(x) dx = δₙₘ - ⌡-∞ + Completeness + + ❙Ψ❭ = ∑ cₙ ❙Eₙ❭ Ψ(x) = ∑ cₙ φₙ(x) + ⁿ ⁿ + + diff --git a/solutions/chap5/prob2 b/solutions/chap5/prob2 index 0d80474..801a25c 100644 --- a/solutions/chap5/prob2 +++ b/solutions/chap5/prob2 @@ -1,17 +1,12 @@ -A particle in an infinite square well has an initial state vector +A particle in an infinite square well has an initial state vector, with A a real number and ι the imaginary unit, ❙Ψ(t=0)❭ = A(❙φ₁❭ - ❙φ₂❭ + ι❙φ₃❭). where ❙φₙ❭ are the energy eigenstates. This also means - ❬Ψ(t=0)❙ = A⃰(❬φ₁❙ - ❬φ₂❙ + ι❬φ₃❙) + ❬Ψ(t=0)❙ = A(❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙) - - - ❙Ψ(t=0)❭ = _͟A͟ (αβ❙φ₁❭ - βγ❙φ₂❭ + αγι❙φ₃❭) - αβγ - In the energy basis, ❙φ₁❭ ≐ ⎛1⎞ ❙φ₂❭ ≐ ⎛0⎞ and ❙φ₃❭ ≐ ⎛0⎞ @@ -23,11 +18,94 @@ So, ⎜-A ⎟ ⎝ιA ⎠. -(𝐚) Multiplying the state vector by its magnitude normalizes it. +(𝐚) The state vector is normalized by taking the quotient of the magnitude. + + ❙Ψ′(t=0)❭ ≐ __͟A͟__ ⎛ 1 ⎞ = _͟1͟ ⎛ 1 ⎞ + √(3A²) ⎜-1 ⎟ √3 ⎜-1 ⎟ + ⎝ ι ⎠ ⎝ ι ⎠. + +When the Hamiltonian operates on ❙φₙ❭ with results according to the general eigenvalue equation, with Eₙ the measured energy of state n, + + Ĥ❙φₙ❭ = Eₙ❙φₙ❭. + +The measured energies for a point particle in an infinite square well are given by, with L the x-width of the well, and m the particle mass, + + Eₙ = n͟²͟π͟²͟ħ͟². + 2mL² + +So, + + Ĥ❙Ψ′(t=0)❭ = _͟1͟ (E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭). + √3 + +(𝐛) There is an equal chance of measuring any of the three values, so 𝓟ₙ=1/3. The measured enemies are given by the previous expression. + + Energy Probability + + π͟²͟ħ͟². 𝓟₁=¹/₃ + 2mL² + + 4͟π͟²͟ħ͟². 𝓟₂=¹/₃ + 2mL² + + 9͟π͟²͟ħ͟². 𝓟₃=¹/₃ + 2mL² + + +The average value of the energy, or the expectation value, is given by + + + │❬Ψ′❙Ĥ❙Ψ′❭│² = │_͟1͟ (❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙)_͟1͟ (E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭)│² + │√3 √3 │. + + ❬Ψ′❙Ĥ❙Ψ′❭ = _͟1͟ (❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙)_͟1͟ (E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭) + √3 √3 + + = ¹/₃ (❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙)(E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭) + + = ¹/₃ (❬φ₁❙E₁❙φ₁❭ - ❬φ₁❙E₂❙φ₂❭ + ❬φ₁❙ιE₃❙φ₃❭) (-❬φ₂❙E₁❙φ₁❭ + ❬φ₂❙E₂❙φ₂❭ - ❬φ₂❙ιE₃❙φ₃❭) + (-ι❬φ₃❙E₁❙φ₁❭ + ι❬φ₃❙E₂❙φ₂❭ - ι❬φ₃❙ιE₃❙φ₃❭). + +Orthogonality is a property of the energy eigenstates: + + ⎧ 0 if n≠m + ❬Eₙ❙Eₘ❭ = δₙₘ, where δₙₘ = ⎨ , so + ⎩ 1 if n=m + +❬Ψ′❙Ĥ❙Ψ′❭ = ¹/₃ (❬φ₁❙E₁❙φ₁❭) (❬φ₂❙E₂❙φ₂❭) (-ι❬φ₃❙ιE₃❙φ₃❭) + += ¹/₃ E₁ E₂ E₃ = ¹/₃ π͟²͟ħ͟² 4͟π͟²͟ħ͟² 9͟π͟²͟ħ͟² = _͟3͟ ⎛π͟²͟ħ͟²⎞³ + 2mL² 2mL² 2mL² 2 ⎝ mL²⎠. + +(𝐜) Therefore, the energy expectation value + + │❬Ψ′❙Ĥ❙Ψ′❭│² = _͟9͟ ⎛π͟²͟ħ͟²⎞⁶ + 4 ⎝ mL²⎠. + +Because the hamiltonian is time-independent, the state vector progresses timewise according to, + + ❙Ψ′(t)❭ = exp(-ι͟Ĥ͟t͟ )❙Ψ′(t=0)❭ = _͟1͟ exp(-ι͟Ĥ͟t͟ ) (❙φ₁❭ - ❙φ₂❭ + ι ❙φ₃❭). + ħ √3 ħ + + _͟1͟ ⎛exp(-ι͟E͟₁͟t͟ )❙φ₁❭ - exp(-ι͟E͟₂͟t͟ )❙φ₂❭ + ι exp(-ι͟E͟₃͟t͟ )❙φ₃❭⎞. + √3 ⎝ ħ ħ ħ ⎠ + + + +The coefficients for the states have the same magnitudes, so the probability of measuring a particular state at time t=0 is 1/3. + + + ❙Ψ(t=0)❭ = _͟A͟ (αβ❙φ₁❭ - βγ❙φ₂❭ + αγι❙φ₃❭) + αβγ + + + + + + + + - ❙Ψ′(t=0)❭ ≐ __͟A͟__ ⎛ 1 ⎞ = _͟1͟ ⎛ 1 ⎞ - √(3A²) ⎜-1 ⎟ √3 ⎜-1 ⎟. - ⎝ ι ⎠ ⎝ ι ⎠ diff --git a/solutions/chap5/prob2.ps b/solutions/chap5/prob2.ps index 70ed598..370dca6 100644 --- a/solutions/chap5/prob2.ps +++ b/solutions/chap5/prob2.ps @@ -1,5 +1,5 @@ %!PS-Adobe-3.0 -%%Title: hello +%%Title: Chap 5 #2 %%Creator: paps version 0.6.7 by Dov Grobgeld %%Pages: (atend) %%BoundingBox: 0 0 595 841 @@ -178,6 +178,81 @@ end /fontdict 1 dict def papsdict begin fontdict begin /AAA { start_ol +3058 3891 m +1732 4386 1262 4930 x +792 5475 792 6395 x +792 7576 1553 8251 x +2315 8928 3646 8928 x +4249 8928 4852 8781 x +5456 8636 6048 8352 x +6048 6696 l +5496 7083 4927 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2730 3909 2893 x 3701 3056 3502 3056 x @@ -716,7 +870,7 @@ end_ol 4824 fwd_x end_ol } def -/XAA { start_ol +/cAA { start_ol 1479 3312 m 3210 3312 l 3423 3312 3423 3152 x @@ -746,7 +900,7 @@ end_ol 4824 fwd_x end_ol } def -/YAA { start_ol +/dAA { start_ol 2559 3312 m 2559 327 l 3784 327 l @@ -769,7 +923,7 @@ end_ol 4824 fwd_x end_ol } def -/ZAA { start_ol +/eAA { start_ol 4263 680 m 4263 609 4127 478 x 3991 347 3771 204 x @@ -799,7 +953,7 @@ end_ol 4824 fwd_x end_ol } def -/aAA { start_ol +/fAA { start_ol 2559 4863 m 2559 327 l 3784 327 l @@ -817,7 +971,7 @@ end_ol 4824 fwd_x end_ol } def -/bAA { start_ol +/gAA { start_ol 4155 1584 m 831 1584 l 919 948 1357 565 x @@ -844,7 +998,7 @@ end_ol 4824 fwd_x end_ol } def -/cAA { start_ol +/hAA { start_ol 2560 3092 m 2415 3092 2282 3064 x 2149 3037 2032 2977 x @@ -885,7 +1039,7 @@ end_ol 4824 fwd_x end_ol } def -/dAA { start_ol +/iAA { start_ol 4125 4464 m 4109 4464 3931 4482 x 3754 4500 3502 4518 x @@ -919,7 +1073,7 @@ end_ol 4824 fwd_x end_ol } def -/eAA { start_ol +/jAA { start_ol 3600 928 m 3600 1227 3334 1389 x 3069 1551 2685 1611 x @@ -961,7 +1115,7 @@ end_ol 4824 fwd_x end_ol } def -/fAA { start_ol +/kAA { start_ol 3672 2658 m 3672 3312 l 4375 3312 l @@ -995,7 +1149,7 @@ end_ol 4824 fwd_x end_ol } def -/gAA { start_ol +/lAA { start_ol 3528 0 m 3528 525 l 2855 -144 2071 -144 x @@ -1023,7 +1177,7 @@ end_ol 4824 fwd_x end_ol } def -/hAA { start_ol +/mAA { start_ol 3524 0 m 3126 0 l 2399 2055 l @@ -1053,7 +1207,7 @@ end_ol 4824 fwd_x end_ol } def -/iAA { start_ol +/nAA { start_ol 3456 2284 m 3456 2641 3190 2866 x 2926 3092 2497 3092 x @@ -1090,7 +1244,7 @@ end_ol 4824 fwd_x end_ol } def -/jAA { start_ol +/oAA { start_ol 2685 0 m 2126 0 l 783 2984 l @@ -1115,7 +1269,7 @@ end_ol 4824 fwd_x end_ol } def -/kAA { start_ol +/pAA { start_ol 2396 3420 m 3162 3420 3688 2896 x 4215 2374 4215 1617 x @@ -1137,7 +1291,214 @@ end_ol 4824 fwd_x end_ol } def -/mAA { start_ol +/qAA { 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 +/rAA { start_ol +903 3312 m +903 2896 l +1099 3182 1285 3300 x +1473 3420 1723 3420 x +2232 3420 2494 2857 x +2940 3420 3411 3420 x +3769 3420 4028 3165 x +4287 2912 4287 2563 x +4287 327 l +4555 327 l +4767 327 4767 167 x +4767 0 4551 0 x +3960 0 l +3960 2530 l +3960 2751 3791 2921 x +3622 3092 3403 3092 x +2997 3092 2559 2445 x +2559 327 l +2827 327 l +3039 327 3039 167 x +3039 0 2823 0 x +2232 0 l +2232 2505 l +2232 2736 2079 2913 x +1927 3092 1723 3092 x +1333 3092 903 2445 x +903 327 l +1144 327 l +1335 327 1335 167 x +1335 0 1125 0 x +274 0 l +72 0 72 167 x +72 327 294 327 x +576 327 l +576 2984 l +294 2984 l +72 2984 72 3150 x +72 3312 294 3312 x +903 3312 l +4824 fwd_x +end_ol + } def +/sAA { start_ol +1119 4863 m +1119 2601 l +1716 3420 2592 3420 x +3333 3420 3846 2911 x +4359 2404 4359 1666 x +4359 913 3841 384 x +3325 -144 2592 -144 x +1692 -144 1119 699 x +1119 0 l +360 0 l +144 0 144 154 x +144 327 360 327 x +792 327 l +792 4536 l +360 4536 l +144 4536 144 4703 x +144 4863 360 4863 x +1119 4863 l +2576 3092 m +1969 3092 1544 2671 x +1119 2252 1119 1650 x +1119 1048 1544 615 x +1969 183 2576 183 x +3172 183 3602 608 x +4032 1033 4032 1626 x +4032 2244 3610 2668 x +3189 3092 2576 3092 x +4824 fwd_x +end_ol + } def +/tAA { start_ol +3999 4863 m +3999 327 l +4383 327 l +4575 327 4575 154 x +4575 0 4375 0 x +3672 0 l +3672 707 l +3107 -144 2224 -144 x +1515 -144 1009 388 x +504 921 504 1650 x +504 2379 1009 2899 x +1515 3420 2224 3420 x +3099 3420 3672 2593 x +3672 4536 l +3288 4536 l +3096 4536 3096 4703 x +3096 4863 3296 4863 x +3999 4863 l +2247 3092 m +1659 3092 1245 2671 x +831 2252 831 1650 x +831 1032 1245 607 x +1659 183 2255 183 x +2844 183 3258 607 x +3672 1032 3672 1634 x +3672 2252 3261 2671 x +2851 3092 2247 3092 x +4824 fwd_x +end_ol + } def +/uAA { 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 +/vAA { start_ol +3528 2650 m +3528 3312 l +4231 3312 l +4431 3312 4431 3150 x +4431 2984 4239 2984 x +3855 2984 l +3855 -247 l +3855 -759 3476 -1135 x +3097 -1512 2552 -1512 x +1652 -1512 l +1440 -1512 1440 -1344 x +1440 -1184 1652 -1184 x +2567 -1184 l +2968 -1184 3247 -894 x +3528 -605 3528 -193 x +3528 844 l +2998 72 2155 72 x +1473 72 988 565 x +504 1059 504 1751 x +504 2443 988 2931 x +1473 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 +/wAA { start_ol +2253 -24 m +751 2984 l +623 2984 l +407 2984 407 3152 x +407 3312 623 3312 x +1542 3312 l +1758 3312 1758 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 +/zAA { start_ol 3008 5592 m 3008 -956 l 1803 -956 l @@ -1146,7 +1507,7 @@ end_ol 4824 fwd_x end_ol } def -/nAA { start_ol +/ABA { start_ol 2823 1440 m 3186 1440 3537 1845 x 3888 2250 3888 2810 x @@ -1201,7 +1562,7 @@ end_ol 4824 fwd_x end_ol } def -/oAA { start_ol +/BBA { start_ol 2376 1941 m 2376 2533 2613 3240 x 2851 3947 3116 4387 x @@ -1222,7 +1583,7 @@ end_ol 4824 fwd_x end_ol } def -/pAA { start_ol +/CBA { start_ol 4171 2664 m 623 2664 l 407 2664 407 2831 x @@ -1240,7 +1601,7 @@ end_ol 4824 fwd_x end_ol } def -/qAA { start_ol +/DBA { start_ol 3855 2829 m 3855 2026 l 3855 1080 3447 486 x @@ -1266,7 +1627,7 @@ end_ol 4824 fwd_x end_ol } def -/rAA { start_ol +/EBA { start_ol 1224 -831 m 1224 -743 1346 -507 x 1470 -272 1620 11 x @@ -1287,7 +1648,7 @@ end_ol 4824 fwd_x end_ol } def -/sAA { start_ol +/FBA { start_ol 1264 -507 m 2720 2407 l 1267 5326 l @@ -1298,7 +1659,7 @@ end_ol 4824 fwd_x end_ol } def -/tAA { start_ol +/GBA { start_ol 4359 1040 m 4359 849 4309 686 x 4261 524 4138 356 x @@ -1338,7 +1699,7 @@ end_ol 4824 fwd_x end_ol } def -/uAA { start_ol +/HBA { start_ol 2543 1364 m 2543 -1155 l 3105 -1155 l @@ -1358,7 +1719,7 @@ end_ol 4824 fwd_x end_ol } def -/vAA { start_ol +/IBA { start_ol 4003 2088 m 790 2088 l 574 2088 574 2255 x @@ -1369,7 +1730,7 @@ end_ol 4824 fwd_x end_ol } def -/wAA { start_ol +/JBA { start_ol 1535 648 m 1535 855 1787 1098 x 2040 1341 2396 1341 x @@ -1397,7 +1758,7 @@ end_ol 4824 fwd_x end_ol } def -/xAA { start_ol +/KBA { start_ol 2559 2088 m 2559 500 l 2559 288 2399 288 x @@ -1418,31 +1779,7 @@ end_ol 4824 fwd_x end_ol } def -/yAA { 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 -/zAA { start_ol +/LBA { start_ol 2460 1084 m 2300 1084 2179 1037 x 2058 991 2009 940 x @@ -1478,7 +1815,7 @@ end_ol 4824 fwd_x end_ol } def -/ABA { start_ol +/MBA { start_ol 2351 939 m 2431 939 l 2662 939 2822 790 x @@ -1493,7 +1830,7 @@ end_ol 4824 fwd_x end_ol } def -/BBA { start_ol +/NBA { start_ol 3528 1487 m 3528 0 l 3096 0 l @@ -1514,73 +1851,7 @@ end_ol 4824 fwd_x end_ol } def -/CBA { start_ol -3528 2650 m -3528 3312 l -4231 3312 l -4431 3312 4431 3150 x -4431 2984 4239 2984 x -3855 2984 l -3855 -247 l -3855 -759 3476 -1135 x -3097 -1512 2552 -1512 x -1652 -1512 l -1440 -1512 1440 -1344 x -1440 -1184 1652 -1184 x -2567 -1184 l -2968 -1184 3247 -894 x -3528 -605 3528 -193 x -3528 844 l -2998 72 2155 72 x -1473 72 988 565 x -504 1059 504 1751 x -504 2443 988 2931 x -1473 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 -/DBA { start_ol -2253 -24 m -751 2984 l -623 2984 l -407 2984 407 3152 x -407 3312 623 3312 x -1542 3312 l -1758 3312 1758 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 -/EBA { start_ol +/OBA { start_ol 2559 327 m 3361 327 l 3567 327 3567 167 x @@ -1606,50 +1877,7 @@ end_ol 4824 fwd_x end_ol } def -/FBA { start_ol -903 3312 m -903 2896 l -1099 3182 1285 3300 x -1473 3420 1723 3420 x -2232 3420 2494 2857 x -2940 3420 3411 3420 x -3769 3420 4028 3165 x -4287 2912 4287 2563 x -4287 327 l -4555 327 l -4767 327 4767 167 x -4767 0 4551 0 x -3960 0 l -3960 2530 l -3960 2751 3791 2921 x -3622 3092 3403 3092 x -2997 3092 2559 2445 x -2559 327 l -2827 327 l -3039 327 3039 167 x -3039 0 2823 0 x -2232 0 l -2232 2505 l -2232 2736 2079 2913 x -1927 3092 1723 3092 x -1333 3092 903 2445 x -903 327 l -1144 327 l -1335 327 1335 167 x -1335 0 1125 0 x -274 0 l -72 0 72 167 x -72 327 294 327 x -576 327 l -576 2984 l -294 2984 l -72 2984 72 3150 x -72 3312 294 3312 x -903 3312 l -4824 fwd_x -end_ol - } def -/GBA { start_ol +/PBA { start_ol 3547 -507 m 2715 -507 l 1264 2407 l @@ -1660,151 +1888,7 @@ end_ol 4824 fwd_x end_ol } def -/HBA { 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 --2279 5652 l --1728 5452 l --1751 6039 l -0 fwd_x -end_ol - } def -/IBA { start_ol -4794 -608 m -4794 -1008 l -0 -1008 l -0 -608 l -4794 -608 l -4824 fwd_x -end_ol - } def -/JBA { start_ol -2418 -536 m -2418 -936 l --2376 -936 l --2376 -536 l -2418 -536 l -0 fwd_x -end_ol - } def -/KBA { start_ol -2247 3092 m -1659 3092 1245 2671 x -831 2252 831 1650 x -831 1032 1245 607 x -1659 183 2255 183 x -2844 183 3258 607 x -3672 1032 3672 1634 x -3672 2252 3261 2671 x -2851 3092 2247 3092 x -3672 3061 m -3672 3279 3831 3279 x -3999 3279 3999 3064 x -3999 327 l -4383 327 l -4575 327 4575 154 x -4575 0 4375 0 x -3672 0 l -3672 707 l -3107 -144 2224 -144 x -1515 -144 1009 388 x -504 921 504 1650 x -504 2379 1009 2899 x -1515 3420 2224 3420 x -3099 3420 3672 2593 x -3672 3061 l -4824 fwd_x -end_ol - } def -/LBA { start_ol -2658 4852 m -3187 4852 3521 4546 x -3855 4241 3855 3733 x -3855 3422 3683 3159 x -3512 2895 3252 2823 x -3805 2635 4046 2333 x -4287 2031 4287 1501 x -4287 732 3772 257 x -3258 -216 2459 -216 x -1990 -216 1471 -16 x -1471 -1184 l -1607 -1184 l -1839 -1184 1839 -1347 x -1839 -1512 1612 -1512 x -700 -1512 l -504 -1512 504 -1347 x -504 -1184 696 -1184 x -1152 -1184 l -1152 3620 l -1152 3838 1204 4027 x -1257 4216 1396 4417 x -1534 4619 1823 4735 x -2112 4852 2527 4852 x -2658 4852 l -2644 4572 m -2102 4572 1791 4374 x -1479 4177 1479 3652 x -1479 338 l -2094 111 2532 111 x -3134 111 3547 517 x -3960 924 3960 1492 x -3960 2079 3534 2371 x -3109 2664 2362 2664 x -2233 2664 2208 2814 x -2232 2831 l -2232 2991 2398 2991 x -2947 2991 3118 3074 x -3289 3157 3432 3417 x -3528 3592 3528 3773 x -3528 4129 3285 4350 x -3042 4572 2644 4572 x -4824 fwd_x -end_ol - } def -/MBA { start_ol -4093 3312 m -4317 3312 4317 3147 x -4317 2984 4133 2984 x -3998 2984 l -2559 -20 l -2559 -1184 l -3067 -1184 l -3279 -1184 3279 -1364 x -3279 -1512 3074 -1512 x -1788 -1512 l -1584 -1512 1584 -1347 x -1584 -1184 1783 -1184 x -2232 -1184 l -2232 -20 l -777 2984 l -681 2984 l -465 2984 465 3147 x -465 3312 681 3312 x -1561 3312 l -1776 3312 1776 3147 x -1776 2984 1561 2984 x -1137 2984 l -2399 337 l -3661 2984 l -3214 2984 l -3015 2984 3015 3147 x -3015 3312 3214 3312 x -4093 3312 l -4824 fwd_x -end_ol - } def -/NBA { start_ol +/QBA { start_ol 2559 4208 m 2559 327 l 3638 327 l @@ -1825,50 +1909,7 @@ end_ol 4824 fwd_x end_ol } def -/OBA { start_ol -1119 4863 m -1119 2601 l -1716 3420 2592 3420 x -3333 3420 3846 2911 x -4359 2404 4359 1666 x -4359 913 3841 384 x -3325 -144 2592 -144 x -1692 -144 1119 699 x -1119 0 l -360 0 l -144 0 144 154 x -144 327 360 327 x -792 327 l -792 4536 l -360 4536 l -144 4536 144 4703 x -144 4863 360 4863 x -1119 4863 l -2576 3092 m -1969 3092 1544 2671 x -1119 2252 1119 1650 x -1119 1048 1544 615 x -1969 183 2576 183 x -3172 183 3602 608 x -4032 1033 4032 1626 x -4032 2244 3610 2668 x -3189 3092 2576 3092 x -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 +/RBA { start_ol 2415 4110 m 2583 4110 2698 3990 x 2814 3871 2814 3703 x @@ -1895,7 +1936,7 @@ end_ol 4824 fwd_x end_ol } def -/RBA { start_ol +/SBA { start_ol 4119 5482 m 3063 4019 3063 2134 x 3063 -1598 l @@ -1909,7 +1950,7 @@ end_ol 4824 fwd_x end_ol } def -/SBA { start_ol +/TBA { start_ol 2559 4927 m 2559 327 l 3645 327 l @@ -1930,7 +1971,7 @@ end_ol 4824 fwd_x end_ol } def -/TBA { start_ol +/UBA { start_ol 704 5511 m 680 5542 680 5607 x 680 5671 731 5718 x @@ -1944,37 +1985,6 @@ end_ol 1760 2162 l 1760 4048 704 5511 x 4824 fwd_x -end_ol - } def -/UBA { start_ol -3999 4863 m -3999 327 l -4383 327 l -4575 327 4575 154 x -4575 0 4375 0 x -3672 0 l -3672 707 l -3107 -144 2224 -144 x -1515 -144 1009 388 x -504 921 504 1650 x -504 2379 1009 2899 x -1515 3420 2224 3420 x -3099 3420 3672 2593 x -3672 4536 l -3288 4536 l -3096 4536 3096 4703 x -3096 4863 3296 4863 x -3999 4863 l -2247 3092 m -1659 3092 1245 2671 x -831 2252 831 1650 x -831 1032 1245 607 x -1659 183 2255 183 x -2844 183 3258 607 x -3672 1032 3672 1634 x -3672 2252 3261 2671 x -2851 3092 2247 3092 x -4824 fwd_x end_ol } def /VBA { start_ol @@ -2112,44 +2122,6 @@ end_ol end_ol } def /bBA { start_ol -2585 1360 m -2221 1360 l -966 4208 l -903 4208 l -903 327 l -1478 327 l -1695 327 1695 167 x -1695 0 1482 0 x -349 0 l -144 0 144 167 x -144 327 335 327 x -576 327 l -576 4208 l -402 4208 l -216 4208 216 4376 x -216 4536 419 4536 x -1164 4536 l -2404 1731 l -3619 4536 l -4371 4536 l -4575 4536 4575 4376 x -4575 4208 4388 4208 x -4215 4208 l -4215 327 l -4456 327 l -4647 327 4647 167 x -4647 0 4441 0 x -3301 0 l -3096 0 3096 167 x -3096 327 3307 327 x -3888 327 l -3888 4208 l -3825 4208 l -2585 1360 l -4824 fwd_x -end_ol - } def -/cBA { start_ol 3783 3312 m 3783 3024 l 1383 327 l @@ -2168,6 +2140,41 @@ end_ol 1008 3312 l 3783 3312 l 4824 fwd_x +end_ol + } def +/cBA { start_ol +1479 1429 m +1479 0 l +720 0 l +504 0 504 167 x +504 327 720 327 x +1152 327 l +1152 4536 l +720 4536 l +504 4536 504 4703 x +504 4863 720 4863 x +1479 4863 l +1479 1819 l +2862 2984 l +2686 2984 l +2470 2984 2470 3152 x +2470 3312 2686 3312 x +3724 3312 l +3940 3312 3940 3152 x +3940 2984 3724 2984 x +3350 2984 l +2094 1945 l +3749 327 l +4108 327 l +4324 327 4324 167 x +4324 0 4108 0 x +3062 0 l +2846 0 2846 167 x +2846 327 3062 327 x +3286 327 l +1847 1739 l +1479 1429 l +4824 fwd_x end_ol } def /dBA { start_ol @@ -2183,6 +2190,24 @@ end_ol end_ol } def /eBA { start_ol +4794 -608 m +4794 -1008 l +0 -1008 l +0 -608 l +4794 -608 l +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 1014 2487 m 1454 786 l 3053 5616 l @@ -2199,7 +2224,7 @@ end_ol 4824 fwd_x end_ol } def -/fBA { start_ol +/hBA { start_ol 970 4285 m 970 4368 1126 4521 x 1282 4674 1625 4822 x @@ -2237,7 +2262,7 @@ end_ol 4824 fwd_x end_ol } def -/gBA { start_ol +/iBA { start_ol 1512 5139 m 1512 5346 1764 5589 x 2017 5832 2372 5832 x @@ -2263,6 +2288,993 @@ end_ol 1591 5004 1551 5043 x 1512 5083 1512 5139 x 4824 fwd_x +end_ol + } def +/jBA { start_ol +984 0 m +495 4208 l +375 4208 l +159 4208 159 4376 x +159 4536 375 4536 x +1567 4536 l +1791 4536 1791 4376 x +1791 4208 1567 4208 x +824 4208 l +1270 376 l +2163 3502 l +2658 3499 l +3539 335 l +3988 4208 l +3220 4208 l +3004 4208 3004 4376 x +3004 4536 3220 4536 x +4414 4536 l +4629 4536 4629 4376 x +4629 4208 4414 4208 x +4294 4208 l +3804 0 l +3292 0 l +2397 3190 l +1478 0 l +984 0 l +4824 fwd_x +end_ol + } def +/kBA { start_ol +3456 2160 m +1335 2160 l +1335 327 l +1719 327 l +1911 327 1911 167 x +1911 0 1706 0 x +644 0 l +432 0 432 167 x +432 327 652 327 x +1008 327 l +1008 4208 l +834 4208 l +648 4208 648 4376 x +648 4536 843 4536 x +1715 4536 l +1911 4536 1911 4376 x +1911 4208 1719 4208 x +1335 4208 l +1335 2487 l +3456 2487 l +3456 4208 l +3072 4208 l +2880 4208 2880 4376 x +2880 4536 3075 4536 x +3947 4536 l +4143 4536 4143 4376 x +4143 4208 3956 4208 x +3783 4208 l +3783 327 l +4146 327 l +4359 327 4359 167 x +4359 0 4154 0 x +3084 0 l +2880 0 2880 167 x +2880 327 3072 327 x +3456 327 l +3456 2160 l +4824 fwd_x +end_ol + } def +/lBA { start_ol +1335 2160 m +1335 327 l +3816 327 l +3816 1266 l +3816 1479 3937 1479 x +4143 1479 4143 1265 x +4143 0 l +642 0 l +432 0 432 167 x +432 327 624 327 x +1008 327 l +1008 4208 l +624 4208 l +432 4208 432 4376 x +432 4536 639 4536 x +3927 4536 l +3927 3478 l +3927 3272 3805 3272 x +3600 3272 3600 3475 x +3600 4208 l +1335 4208 l +1335 2487 l +2448 2487 l +2448 2802 l +2448 2991 2615 2991 x +2775 2991 2775 2796 x +2775 1850 l +2775 1656 2615 1656 x +2448 1656 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4314 x +2925 4644 2448 4644 x +2003 4644 1705 4286 x +1407 3929 1407 3409 x +1407 2930 1705 2580 x +2003 2232 2416 2232 x +2615 2232 2810 2328 x +3004 2426 3124 2527 x +3243 2630 3381 2828 x +3521 3027 3556 3092 x +3592 3157 3679 3328 x +4824 fwd_x +end_ol + } def +/ACA { start_ol +2232 6393 m +2559 6393 l +2559 -1598 l +2232 -1598 l +2232 6393 l +4824 fwd_x +end_ol + } def +/BCA { start_ol +2396 4644 m +3232 4644 3795 3944 x +4359 3244 4359 2217 x +4359 1238 3783 546 x +3208 -144 2396 -144 x +1576 -144 1003 550 x +432 1245 432 2250 x +432 3254 1003 3948 x +1576 4644 2396 4644 x +2395 4316 m +1719 4316 1238 3708 x +759 3100 759 2250 x +759 1399 1238 790 x +1719 183 2395 183 x +3064 183 3548 786 x +4032 1390 4032 2225 x +4032 3100 3556 3708 x +3080 4316 2395 4316 x +4824 fwd_x +end_ol + } def +/CCA { start_ol +2351 939 m +2431 939 l +2662 939 2822 790 x +2982 643 2982 419 x +2982 195 2822 43 x +2662 -108 2431 -108 x +2351 -108 l +2119 -108 1959 43 x +1800 195 1800 419 x +1800 643 1959 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