From 563ee4f2f803b41539853e92aef8622bc6b5e528 Mon Sep 17 00:00:00 2001 From: caes Date: Sat, 26 Mar 2016 18:02:07 -0400 Subject: [PATCH] finished most of chap 7 (a) homework --- solutions/chap7/lectureprob.ps | 2186 ----------------- solutions/chap7/prob5 | 107 +- solutions/chap7/prob5.ps | 3448 --------------------------- solutions/chap7/prob7 | 3 - solutions/chap7/prob7.ps | 3995 -------------------------------- solutions/chap7/prob8 | 287 ++- solutions/chap7/prob8.ps | 2452 -------------------- 7 files changed, 234 insertions(+), 12244 deletions(-) delete mode 100644 solutions/chap7/lectureprob.ps delete mode 100644 solutions/chap7/prob5.ps delete mode 100644 solutions/chap7/prob7.ps delete mode 100644 solutions/chap7/prob8.ps diff --git a/solutions/chap7/lectureprob.ps b/solutions/chap7/lectureprob.ps deleted file mode 100644 index 57bb532..0000000 --- a/solutions/chap7/lectureprob.ps +++ /dev/null @@ -1,2186 +0,0 @@ -%!PS-Adobe-3.0 -%%Title: Otho Ulrich: Class Prob -%%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. 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} def -/IBA { start_ol -2232 -680 m -2232 4536 l -1499 4536 l -1296 4536 1296 4695 x -1296 4863 1503 4863 x -2559 4863 l -2559 -1008 l -1503 -1008 l -1296 -1008 1296 -848 x -1296 -680 1499 -680 x -2232 -680 l -4824 fwd_x -end_ol - } def -/JBA { start_ol -4171 2664 m -623 2664 l -407 2664 407 2831 x -407 2991 623 2991 x -4171 2991 l -4387 2991 4387 2831 x -4387 2664 4171 2664 x -4171 1512 m -623 1512 l -407 1512 407 1679 x -407 1839 623 1839 x -4171 1839 l -4387 1839 4387 1679 x -4387 1512 4171 1512 x -4824 fwd_x -end_ol - } def -/KBA { start_ol -4003 2088 m -790 2088 l -574 2088 574 2255 x -574 2415 790 2415 x -4003 2415 l -4219 2415 4219 2255 x -4219 2088 4003 2088 x -4824 fwd_x -end_ol - } def -/LBA { start_ol -3999 4208 m -3999 1481 l -3999 794 3535 325 x -3073 -144 2396 -144 x -1711 -144 1251 320 x -792 786 792 1481 x -792 4208 l -551 4208 l -360 4208 360 4376 x -360 4536 565 4536 x -1705 4536 l -1911 4536 1911 4376 x -1911 4208 1699 4208 x -1119 4208 l -1119 1479 l -1119 928 1486 555 x -1854 183 2395 183 x -2928 183 3299 560 x -3672 937 3672 1479 x -3672 4208 l -3091 4208 l -2880 4208 2880 4376 x -2880 4536 3085 4536 x -4225 4536 l -4431 4536 4431 4376 x -4431 4208 4240 4208 x -3999 4208 l -4824 fwd_x -end_ol - } def -/MBA { 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 -/NBA { 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 -/OBA { start_ol -3456 2284 m -3456 2641 3190 2866 x -2926 3092 2497 3092 x -2170 3092 1955 2962 x -1740 2831 1420 2451 x -1335 2347 l -1335 327 l -1690 327 l -1911 327 1911 167 x -1911 0 1689 0 x -645 0 l -432 0 432 167 x -432 327 644 327 x -1008 327 l -1008 3776 l -624 3776 l -432 3776 432 3939 x -432 4104 624 4104 x -1008 4104 l -1008 4536 l -624 4536 l -432 4536 432 4703 x -432 4863 632 4863 x -1335 4863 l -1335 4104 l -2423 4104 l -2631 4104 2631 3939 x -2631 3776 2423 3776 x -1335 3776 l -1335 2751 l -1631 3117 1903 3268 x -2176 3420 2535 3420 x -3084 3420 3433 3106 x -3783 2793 3783 2309 x -3783 327 l -4138 327 l -4359 327 4359 167 x -4359 0 4147 0 x -3156 0 l -2952 0 2952 167 x -2952 327 3141 327 x -3456 327 l -3456 2284 l -4824 fwd_x -end_ol - } def -/PBA { start_ol -1733 4971 m -2725 4971 3398 4217 x -4071 3465 4071 2264 x -4071 1218 3621 554 x -3171 -108 2457 -108 x -1876 -108 1478 338 x -1080 786 1080 1433 x -1080 2040 1473 2479 x -1867 2919 2408 2919 x -3243 2919 3728 1945 x -3744 2248 3744 2288 x -3744 2756 3628 3143 x -3513 3530 3322 3772 x -3132 4015 2913 4196 x -2695 4377 2460 4469 x -2226 4563 2055 4603 x -1884 4644 1765 4644 x -1542 4644 1410 4608 x -1279 4572 1255 4572 x -1191 4572 1147 4614 x -1103 4658 1103 4717 x -1103 4818 1290 4894 x -1478 4971 1733 4971 x -3679 1513 m -3576 1705 3508 1813 x -3441 1921 3270 2140 x -3099 2360 2881 2476 x -2662 2592 2416 2592 x -2003 2592 1705 2247 x -1407 1904 1407 1433 x -1407 922 1705 570 x -2003 219 2448 219 x -2925 219 3218 542 x -3513 866 3679 1513 x -4824 fwd_x -end_ol - } def -/QBA { start_ol -3816 5059 m -1263 -504 l -1193 -646 1092 -646 x -1029 -646 982 -598 x -936 -551 936 -495 x -936 -455 975 -367 x -3528 5195 l -3589 5338 3699 5338 x -3762 5338 3808 5289 x -3855 5242 3855 5186 x -3855 5139 3816 5059 x -4824 fwd_x -end_ol - } def -/RBA { start_ol -3855 2829 m -3855 2026 l -3855 1080 3447 486 x -3039 -108 2395 -108 x -1751 -108 1343 486 x -936 1080 936 2026 x -936 2829 l -936 3783 1343 4377 x -1751 4971 2395 4971 x -3039 4971 3447 4377 x -3855 3783 3855 2829 x -2395 219 m -2922 219 3225 779 x -3528 1339 3528 2109 x -3528 2264 l -1263 2264 l -1263 2109 l -1263 1307 1577 762 x -1891 219 2395 219 x -2395 4644 m -1868 4644 1566 4097 x -1263 3550 1263 2800 x -1263 2592 l -3528 2592 l -3528 2800 l -3528 3582 3214 4113 x -2900 4644 2395 4644 x -4824 fwd_x -end_ol - } def -/SBA { start_ol -4125 4464 m -4109 4464 3931 4482 x -3754 4500 3502 4518 x -3251 4536 3057 4536 x -2678 4536 2439 4339 x -2199 4142 2199 3823 x -2199 3312 l -3648 3312 l -3855 3312 3855 3152 x -3855 2984 3648 2984 x -2199 2984 l -2199 327 l -3575 327 l -3783 327 3783 167 x -3783 0 3576 0 x -1143 0 l -936 0 936 167 x -936 327 1138 327 x -1872 327 l -1872 2984 l -1210 2984 l -1008 2984 1008 3152 x -1008 3312 1210 3312 x -1872 3312 l -1872 3818 l -1872 4273 2209 4568 x -2548 4863 3067 4863 x -3611 4863 4147 4765 x -4287 4727 4287 4614 x -4287 4464 4125 4464 x -4824 fwd_x -end_ol - } def -/TBA { 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 -/UBA { start_ol -3672 2658 m -3672 3312 l -4375 3312 l -4575 3312 4575 3150 x -4575 2984 4383 2984 x -3999 2984 l -3999 -1184 l -4383 -1184 l -4575 -1184 4575 -1344 x -4575 -1512 4374 -1512 x -2937 -1512 l -2736 -1512 2736 -1344 x -2736 -1184 2938 -1184 x -3672 -1184 l -3672 837 l -3138 72 2239 72 x -1507 72 1005 558 x -504 1044 504 1751 x -504 2459 1005 2939 x -1507 3420 2247 3420 x -3130 3420 3672 2658 x -2247 3092 m -1651 3092 1240 2704 x -831 2316 831 1751 x -831 1186 1240 793 x -1651 399 2247 399 x -2836 399 3253 788 x -3672 1179 3672 1735 x -3672 2315 3261 2703 x -2851 3092 2247 3092 x -4824 fwd_x -end_ol - } def -/VBA { start_ol -2559 4208 m -2559 327 l -3638 327 l -3855 327 3855 167 x -3855 0 3637 0 x -1146 0 l -936 0 936 167 x -936 327 1146 327 x -2232 327 l -2232 4208 l -1146 4208 l -936 4208 936 4376 x -936 4536 1146 4536 x -3637 4536 l -3855 4536 3855 4376 x -3855 4208 3638 4208 x -2559 4208 l -4824 fwd_x -end_ol - } def -/WBA { 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 2448 1845 x -2448 2160 l -1335 2160 l -4824 fwd_x -end_ol - } def -/XBA { start_ol -3855 2829 m -3855 2026 l -3855 1080 3447 486 x -3039 -108 2395 -108 x -1751 -108 1343 486 x -936 1080 936 2026 x -936 2829 l -936 3783 1343 4377 x -1751 4971 2395 4971 x -3039 4971 3447 4377 x -3855 3783 3855 2829 x -2395 4644 m -1868 4644 1566 4093 x -1263 3543 1263 2788 x -1263 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 - 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The ⎧ b = √2 a ⎨ (a + c) = √2 b ⎩ b = √2 c - + +Following this to conclusion just like with spin operators will provide the eigenstates, and then from that the wave function can be expressed using the x basis, and probabilities obtained. +I need to stop here, but I will produce at least sthe histogram from part a: +(𝐜) +𝓟(L̂𝓏) - - - - - - - - - - - -Applying the operator to the states in Ψ, - - L̂𝓍❙1 1❭ ≐ - ħ͟ ⎛ 0 1 0 ⎞⎛1⎞ = ħ͟ ⎛0⎞ = ħ͟ ❙1 0❭. - √2 ⎜ 1 0 1 ⎟⎜0⎟ √2 ⎜1⎟ √2 - ⎝ 0 1 0 ⎠⎝0⎠ ⎝0⎠ - - L̂𝓍❙1 0❭ ≐ - ħ͟ ⎛ 0 1 0 ⎞⎛0⎞ = ħ͟ ⎛1⎞ = ħ͟ (❙1 1❭ + ❙1 -1❭), and - √2 ⎜ 1 0 1 ⎟⎜1⎟ √2 ⎜0⎟ √2 - ⎝ 0 1 0 ⎠⎝0⎠ ⎝1⎠ - - L̂𝓍❙1 -1❭ ≐ - ħ͟ ⎛ 0 1 0 ⎞⎛0⎞ = ħ͟ ⎛0⎞ = ħ͟ ❙1 0❭. - √2 ⎜ 1 0 1 ⎟⎜0⎟ √2 ⎜1⎟ √2 - ⎝ 0 1 0 ⎠⎝1⎠ ⎝0⎠ - - - L̂𝓍❙Ψ❭ = ⎛ 2͟ L̂𝓍❙1 1❭ + ι 3͟ L̂𝓍❙1 0❭ - 4͟ L̂𝓍❙1 -1❭ ⎞ - ⎝ √29 √29 √29 ⎠ - - 2͟ L̂𝓍❙1 1❭ = 2͟ ħ❙1 0❭, - √29 √58 - - ι 3͟ L̂𝓍❙1 0❭ = ι 3͟ ħ (❙1 1❭ + ❙1 -1❭), and - √29 √58 - - 4͟ L̂𝓍❙1 -1❭ = 4͟ ħ❙1 0❭. - √29 √58 - -Then, - - L̂𝓍❙Ψ❭ = ħ ⎛ -2͟ ❙1 0❭ + ι 3͟ (❙1 1❭ + ❙1 -1❭)⎞ - ⎝ √58 √58 ⎠ - -Normalizing the function, - - C⎛⎛-2͟ ⎞² + ⎛ι 3͟ ⎞² + ⎛ι 3͟ ⎞²⎞ = 1. - ⎝⎝√58⎠ ⎝ √58⎠ ⎝ √58⎠ ⎠ - - C⎛4 - 9 - 9⎞ = 58 = C(14). - ⎝ ⎠ - - C = 58/14 = 29/7. - -The probability of measuring one of the possible angular momenta will be given by - - │❬l m❙L̂𝓍❙Ψ❭│². - - ❬1 1❙L̂𝓍❙Ψ❭ = ❬1 1❙2͟9͟⎛ -2͟ ❙1 0❭ + ι 3͟ (❙1 1❭ + ❙1 -1❭)⎞ - 7 ⎝ √58 √58 ⎠ - - = 2͟9͟ ι͟3͟ ❬1 1❙1 1❭ = ι 8͟7͟ - 7 √58 7√58 - - - - - - L̂𝓍❙Ψ❭ = ⎛ + ι 3͟ L̂𝓍❙1 0❭ - 4͟ L̂𝓍❙1 -1❭ ⎞ - ⎝ √29 √29 ⎠ - - + ╭─────────────────────────╮ + │ │ + │ │ + │ │ + │ │ + │ │ + │                      │  +    │                         │     + ¹⁶/₂₉ ├   ▓                     │     +  │   ▓                     │     +  │   ▓                     │  +  │   ▓                     │  +  │   ▓                     │  +  │   ▓                     │  + │   ▓                     │  + ⁹/₂₉ ├   ▓        ▓            │  +  │   ▓        ▓            │  +  │   ▓        ▓            │  +  │   ▓        ▓            │  +  │   ▓        ▓            │  + ⁴/₂₉ ├   ▓        ▓        ▓   │  +  │   ▓        ▓        ▓   │  +  │   ▓        ▓        ▓   │  +  │   ▓        ▓        ▓   │  +    ╰─────────────────────────╯ + -ħ 0 ħ + diff --git a/solutions/chap7/prob5.ps b/solutions/chap7/prob5.ps deleted file mode 100644 index 1f9f504..0000000 --- a/solutions/chap7/prob5.ps +++ /dev/null @@ -1,3448 +0,0 @@ -%!PS-Adobe-3.0 -%%Title: Otho Ulrich: Chap 7 #5 -%%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! 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start_ol -2418 -536 m -2418 -936 l --2376 -936 l --2376 -536 l -2418 -536 l -0 fwd_x -end_ol - } def -/DBA { 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 -/EBA { 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 -/FBA { 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 -/GBA { start_ol -970 4285 m -970 4368 1126 4521 x -1282 4674 1625 4822 x -1969 4971 2377 4971 x -2976 4971 3379 4614 x -3783 4258 3783 3737 x -3783 3272 3523 3000 x -3263 2728 2959 2648 x -3431 2454 3714 2112 x -3999 1771 3999 1361 x -3999 751 3507 321 x -3016 -108 2328 -108 x -1857 -108 1317 128 x -778 364 778 564 x -778 621 825 669 x -874 717 930 717 x -986 717 1154 592 x -1321 468 1633 343 x -1945 219 2336 219 x -2880 219 3276 556 x -3672 895 3672 1361 x -3672 1836 3245 2178 x -2820 2520 2233 2520 x -2016 2520 2016 2687 x -2016 2847 2227 2847 x -3456 2847 3456 3729 x -3456 4123 3144 4383 x -2832 4644 2369 4644 x -2034 4644 1797 4566 x -1562 4488 1470 4392 x -1378 4295 1290 4217 x -1202 4140 1130 4140 x -1066 4140 1018 4181 x -970 4224 970 4285 x -4824 fwd_x -end_ol - } def -/HBA { start_ol -3855 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fwd_x -end_ol - } def -/KBA { 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 -/LBA { start_ol -1255 291 m -1279 291 1410 255 x -1542 219 1765 219 x -2083 219 2416 365 x -2750 513 3052 783 x -3354 1054 3549 1514 x -3744 1975 3744 2540 x -3744 2580 3728 2890 x -3243 1904 2408 1904 x -1867 1904 1473 2349 x -1080 2794 1080 3409 x -1080 4065 1478 4518 x -1876 4971 2457 4971 x -3171 4971 3621 4298 x -4071 3625 4071 2565 x -4071 1379 3394 635 x -2718 -108 1733 -108 x -1478 -108 1290 -31 x -1103 45 1103 146 x -1103 204 1147 247 x -1191 291 1255 291 x -3679 3328 m -3513 3985 3218 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 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x -2936 4900 2936 5027 x -2936 5234 2767 5388 x -2598 5544 2371 5544 x -2167 5544 2007 5428 x -1847 5313 1792 5131 x -1760 5004 1648 5004 x -1591 5004 1551 5043 x -1512 5083 1512 5139 x -4824 fwd_x -end_ol - } def -/fBA { start_ol -1263 4160 m -1263 3903 1571 3735 x -1879 3567 2358 3567 x -3151 3567 3750 2889 x -4143 2454 4143 1743 x -4143 1011 3634 469 x -3127 -72 2343 -72 x -1613 -72 1094 469 x -576 1011 576 1743 x -576 2488 949 2876 x -1402 3336 1657 3407 x -1359 3454 1147 3656 x -936 3858 936 4143 x -936 4595 1212 4837 x -1489 5079 1949 5079 x -2365 5079 3600 4759 x -3600 4944 l -3600 5151 3816 5151 x -3927 5151 3927 4939 x -3927 4330 l -3927 4104 3816 4104 x -3600 4104 3600 4333 x -3600 4443 l -2412 4752 1973 4752 x -1649 4752 1455 4600 x -1263 4448 1263 4160 x -2343 255 m -2999 255 3407 696 x -3816 1138 3816 1743 x -3816 2077 3695 2352 x -3576 2626 3399 2785 x -3224 2945 3011 3052 x -2800 3160 2631 3199 x -2463 3240 2343 3240 x -1732 3240 1317 2805 x -903 2372 903 1743 x -903 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100644 --- a/solutions/chap7/prob8 +++ b/solutions/chap7/prob8 @@ -1,4 +1,4 @@ -7.35 is nothing more than a definition of spherical coordinates. +The transformation from polar coordinates to cartesian is the set of equations ⎧ x = r sinθ cosϕ ⎪ @@ -6,21 +6,142 @@ ⎪ ⎩ z = r cosθ +The transformation from cartesian to polar coordinates is the set + +⎧ r = √(x²+y²+z²) +⎪ +⎪ cos(θ) = z͟ = _͟z͟ +⎨ r √(x²+y²+z²) +⎪ +⎪ tan(ϕ) = y͟ +⎩ x + Some differential forms may come in handy. +⎧ dcos(θ) = -sin(θ) dθ +⎨ +⎪ dtan(ϕ) = _͟1͟ dϕ +⎩ cos²(ϕ) + ∂/∂θ: -⎧ ∂x = r cosϕ cosθ ∂θ + +⎧ ∂͟x = r cosϕ cosθ +⎪ ∂θ ⎪ -⎨ ∂y = r sinϕ cosθ ∂θ +⎨ ∂͟y = r sinϕ cosθ +⎪ ∂θ ⎪ -⎩ ∂z = - r sinθ ∂θ +⎪ ∂͟z = -r sinθ +⎩ ∂θ ∂/∂ϕ: -⎧ ∂x = - r sinθ sinϕ ∂ϕ + +⎧ ∂͟x = -r sinθ sinϕ +⎪ ∂ϕ ⎪ -⎨ ∂y = r sinθ cosϕ ∂ϕ +⎨ ∂͟y = r sinθ cosϕ +⎪ ∂ϕ +⎪ +⎪ ∂͟z = 0 +⎩ ∂ϕ + +∂/∂x: + +⎧ ∂͟r = √(x²+y²+z²) = x͟ +⎪ ∂x r ⎪ -⎩ ∂z = 0 ∂ϕ +⎪ ∂͟cos(θ) = -z͟x͟ +⎨ ∂x r³ +⎪ +⎪ ∂͟tan(ϕ) = -y͟ +⎩ ∂x x² + +∂/∂y: + +⎧ ∂͟r = √(x²+y²+z²) = y͟ +⎪ ∂y r +⎪ +⎪ ∂͟cos(θ) = -z͟y͟ +⎨ ∂y r³ +⎪ +⎪ ∂͟tan(ϕ) = 1͟ +⎩ ∂y x + +∂/∂z: + +⎧ ∂͟r = √(x²+y²+z²) = z͟ +⎪ ∂z r +⎪ +⎪ ∂͟cos(θ) = z͟ = 1͟ ⎛r ∂͟z͟ - z ∂͟r͟⎞ = 1͟ - z͟² +⎨ ∂z r r² ⎝ ∂z ∂z⎠ r r³ +⎪ +⎪ ∂͟tan(ϕ) = 0 +⎩ ∂z + + +These differential forms can be used to transform each cartesian differentiation operator. + + ∂͟ = ∂͟r͟ ∂͟ + ∂͟c͟o͟s͟θ͟ ∂͟ + ∂͟t͟a͟n͟ϕ͟ ∂͟ ; + ∂x ∂x ∂r ∂x ∂cosθ ∂x ∂tanϕ + + ∂͟ = x͟ ∂͟ - z͟x͟ _͟1͟ ∂͟ - y͟ cos²(ϕ) ∂͟ ; + ∂x r ∂r r³ -sinθ ∂θ x² ∂ϕ + + ∂͟ = sin(θ) cos(ϕ) ∂͟ + ∂x ∂r + + 1͟ sin(θ) cos(ϕ) cos(θ) _͟1͟ ∂͟ + r sin(θ) ∂θ + - r͟ s͟i͟n͟(ϕ͟) s͟i͟n͟(θ͟) cos²(ϕ͟) ∂͟ ; + r² cos²(ϕ) sin²(θ) ∂ϕ + + ∂͟ = sin(θ) cos(ϕ) ∂͟ + 1͟ cos(ϕ) cos(θ) ∂͟ - 1͟ s͟i͟n͟(ϕ͟) ∂͟ . + ∂x ∂r r ∂θ r sin(θ) ∂ϕ + + ─────────────────────────────────────────────── + + ∂͟ = ∂͟r͟ ∂͟ + ∂͟c͟o͟s͟θ͟ ∂͟ + ∂͟t͟a͟n͟ϕ͟ ∂͟ ; + ∂y ∂y ∂r ∂y ∂cosθ ∂y ∂tanϕ + + ∂͟ = y͟ ∂͟ - z͟y͟ ∂͟ + 1͟ ∂͟ ; + ∂y r ∂r r³ ∂cosθ x ∂tanϕ + + ∂͟ = sin(θ) sin(ϕ) ∂͟ - 1͟ sin(θ) cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ ; + ∂y ∂r r -sin(θ) ∂θ r sin(θ) ∂ϕ + + ∂͟ = sin(θ) sin(ϕ) ∂͟ + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ . + ∂y ∂r r ∂θ r sin(θ) ∂ϕ + + ─────────────────────────────────────────────── + + ∂͟ = ∂͟r͟ ∂͟ + ∂͟c͟o͟s͟θ͟ ∂͟ + ∂͟t͟a͟n͟ϕ͟ ∂͟ ; + ∂z ∂z ∂r ∂z ∂cosθ ∂z ∂tanϕ + + ∂͟ = z͟ ∂͟ + ⎛1͟ - z͟²⎞∂͟ + 0 ; + ∂z r ∂r ⎝r r³⎠∂cosθ + + ∂͟ = cos(θ) ∂͟ + 1͟⎛1 - cos²(θ)⎞ _͟1͟ ∂͟ ; + ∂z ∂r r⎝ ⎠ -sin(θ) ∂θ + + ∂͟ = cos(θ) ∂͟ - _͟1͟_ ⎛1 - cos²(θ)⎞ ∂͟ ; + ∂z ∂r rsin(θ)⎝ ⎠ ∂θ + + ∂͟ = cos(θ) ∂͟ - _͟1͟_ sin²(θ) ∂͟ ; + ∂z ∂r rsin(θ) ∂θ + + ∂͟ = cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ . + ∂z ∂r r ∂θ + + +The set of differential operator transformations is + +⎧ ∂͟ = sin(θ) cos(ϕ) ∂͟ + 1͟ cos(ϕ) cos(θ) ∂͟ - 1͟ s͟i͟n͟(ϕ͟) ∂͟ +⎪ ∂x ∂r r ∂θ r sin(θ) ∂ϕ +⎪ +⎨ ∂͟ = sin(θ) sin(ϕ) ∂͟ + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ +⎪ ∂y ∂r r ∂θ r sin(θ) ∂ϕ +⎪ +⎪ ∂͟ = cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ +⎩ ∂z ∂r r ∂θ 7.47 is the set of algebraic conditions expressed by the vector definition 𝐋 = 𝐫 × 𝐩. @@ -45,91 +166,87 @@ Substituting 7.35 into 7.47, ⎪ L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟_ - r sinθ sinϕ ∂͟_ ) ⎩ ∂y ∂x +Substituting the transformed differentiation operators, - -Geometry is shown on the attached notes page. - -For L̂𝓍: - - ∂͟z͟ = -r sinθ - ∂θ - ∂͟y͟ = r sinθ cosϕ - ∂ϕ - - L̂𝓍 = -ιħ ( r sinθ sinϕ ∂͟θ͟ ∂͟_ - r cosθ ∂͟ϕ͟ ∂͟_ ) - ∂θ ∂z ∂ϕ ∂y - - L̂𝓍 = ιħ ( -r sinθ sinϕ ∂͟θ͟ ∂͟_ + r cosθ ∂͟ϕ͟ ∂͟_ ) - ∂z ∂θ ∂y ∂ϕ - - L̂𝓍 = ιħ ( −͟r͟ s͟i͟n͟θ͟ sinϕ ∂͟_ + c͟o͟s͟θ͟ ∂͟_ ) - -r sinθ ∂θ sinθ cosϕ ∂ϕ - - L̂𝓍 = ιħ ( sinϕ ∂͟_ + c͟o͟t͟θ͟ ∂͟_ ) - ∂θ cosϕ ∂ϕ - -For L̂𝓎: - - ∂x = r cosϕ cosθ ∂θ - ∂z = 0 ∂ϕ - - L̂𝓎 = -ιħ (r cosθ ∂͟_ - r sinθ cosϕ ∂͟_ ) - ∂x ∂z - - L̂𝓎 = ιħ (-r cosθ ∂͟θ͟ ∂͟_ + r sinθ cosϕ ∂͟ϕ͟ ∂͟_ ) - ∂θ ∂x ∂ϕ ∂z - - L̂𝓎 = ιħ (-r cosθ ∂͟θ͟ ∂͟_ + r sinθ cosϕ ∂͟ϕ͟ ∂͟_ ) - ∂x ∂θ ∂z ∂ϕ - - L̂𝓎 = ιħ (- _͟1͟ cosθ ∂͟_ + r sinθ cosϕ 0 ∂͟_ ) - cosϕ cosθ ∂θ ∂ϕ - - L̂𝓎 = ιħ (- _͟1͟ ∂͟_ ) - cosϕ ∂θ - - L̂𝓎 = -ιħ _͟1͟ ∂͟_ - cosϕ ∂θ - -For L̂𝓏: - - ∂x = -r sinθ sinϕ ∂ϕ - ∂y = r sinθ cosϕ ∂ϕ - - L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟͟ϕ͟ ∂͟_ - r sinθ sinϕ ∂͟͟ϕ͟ ∂͟_ ) - ∂ϕ ∂y ∂ϕ ∂x - - L̂𝓏 = -ιħ (r͟ s͟i͟n͟θ͟ c͟o͟s͟ϕ͟ ∂͟_ + r͟ s͟i͟n͟θ͟ s͟i͟n͟ϕ͟ ∂͟_ ) - r sinθ cosϕ ∂ϕ r sinθ sinϕ ∂ϕ - - L̂𝓏 = -ιħ ∂͟_ ( 1 + 1 ) - ∂ϕ - - L̂𝓏 = -2ιħ ∂͟_ - ∂ϕ - -So, according to my calculus, the final solutions should be the set - -⎧ L̂𝓍 = ιħ ( sinϕ ∂͟_ + c͟o͟t͟θ͟ ∂͟_ ) -⎪ ∂θ cosϕ ∂ϕ +⎧ L̂𝓍 = -ιħ⎛r sinθ sinϕ ⎛cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ ⎞ +⎪ ⎝ ⎝ ∂r r ∂θ⎠ +⎪ +⎪ - r cosθ ⎛sin(θ) sin(ϕ) ∂͟ + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ ⎞⎞ +⎪ ⎝ ∂r r ∂θ r sin(θ) ∂ϕ⎠⎠ ⎪ -⎨ L̂𝓎 = -ιħ _͟1͟ ∂͟_ -⎪ cosϕ ∂θ ⎪ -⎪ L̂𝓏 = -2ιħ ∂͟_ -⎩ ∂ϕ - +⎪ L̂𝓎 = -ιħ⎛r cosθ⎛sin(θ) cos(ϕ) ∂͟ + 1͟ cos(ϕ) cos(θ) ∂͟ - 1͟ s͟i͟n͟(ϕ͟) ∂͟ ⎞ +⎪ ⎝ ⎝ ∂r r ∂θ r sin(θ) ∂ϕ⎠ +⎨ +⎪ - r sinθ cosϕ ⎛cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ ⎞⎞ +⎪ ⎝ ∂r r ∂θ⎠⎠ +⎪ +⎪ +⎪ L̂𝓏 = -ιħ⎛r sinθ cosϕ⎛sin(θ) sin(ϕ) ∂͟ + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ ⎞ +⎪ ⎝ ⎝ ∂r r ∂θ r sin(θ) ∂ϕ⎠ +⎪ +⎪ - r sinθ sinϕ ⎛sin(θ) cos(ϕ) ∂͟ + 1͟ cos(ϕ) cos(θ) ∂͟ - 1͟ s͟i͟n͟(ϕ͟) ∂͟ ⎞⎞ +⎩ ⎝ ∂r r ∂θ r sin(θ) ∂ϕ⎠⎠ -The spherical representation, i.e. ending place, is the set +Simplifying... + +⎧ L̂𝓍 = -ιħ⎛r sinθ sinϕ cosθ ∂͟ - sinϕ sin²θ ∂͟ +⎪ ⎝ ∂r ∂θ +⎪ +⎪ - r cosθ sinθ sinϕ ∂͟ - cos²θ sinϕ ∂͟ - c͟o͟s͟θ͟ cosϕ ∂͟ ⎞ +⎪ ∂r ∂θ sinθ ∂ϕ⎠ +⎪ +⎪ +⎪ L̂𝓎 = -ιħ⎛r cosθ sinθ cosϕ ∂͟ + cosθ cosϕ cosθ ∂͟ - c͟o͟s͟θ͟ sinϕ ∂͟ +⎪ ⎝ ∂r ∂θ sinθ ∂ϕ +⎨ +⎪ - r sinθ cosϕ cosθ ∂͟ + sinθ cosϕ sinθ ∂͟ ⎞ +⎪ ∂r ∂θ⎠ +⎪ +⎪ +⎪ L̂𝓏 = -ιħ⎛r sin²θ cosϕ sinϕ ∂͟ + sinθ cosθ sinϕ cosϕ ∂͟ + cosϕ c͟o͟s͟ϕ͟ ∂͟ +⎪ ⎝ ∂r ∂θ ∂ϕ +⎪ +⎪ - r sin²θ sinϕ cosϕ ∂͟ - sinθ cosθ sinϕ cosϕ ∂͟ + sinθ sinϕ s͟i͟n͟ϕ͟ ∂͟ ⎞ +⎩ ∂r ∂θ sinθ ∂ϕ⎠ + + +⎧ L̂𝓍 = -ιħ⎛- sinϕ sin²θ ∂͟ - cos²θ sinϕ ∂͟ - c͟o͟s͟θ͟ cosϕ ∂͟ ⎞ +⎪ ⎝ ∂θ ∂θ sinθ ∂ϕ⎠ +⎪ +⎨ L̂𝓎 = -ιħ⎛cosθ cosϕ cosθ ∂͟ - c͟o͟s͟θ͟ sinϕ ∂͟ + sinθ cosϕ sinθ ∂͟ ⎞ +⎪ ⎝ ∂θ sinθ ∂ϕ ∂θ⎠ +⎪ +⎪ L̂𝓏 = -ιħ⎛cosϕ c͟o͟s͟ϕ͟ ∂͟ + sinθ sinϕ s͟i͟n͟ϕ͟ ∂͟ ⎞ +⎩ ⎝ ∂ϕ sinθ ∂ϕ⎠ + + +⎧ L̂𝓍 = ιħ⎛ sinϕ sin²θ ∂͟ + cos²θ sinϕ ∂͟ + c͟o͟s͟θ͟ cosϕ ∂͟ ⎞ +⎪ ⎝ ∂θ ∂θ sinθ ∂ϕ⎠ +⎪ +⎨ L̂𝓎 = ιħ⎛-cos²θ cosϕ ∂͟ - sin²θ cosϕ ∂͟ + c͟o͟s͟θ͟ sinϕ ∂͟ ⎞ +⎪ ⎝ ∂θ ∂θ sinθ ∂ϕ ⎠ +⎪ +⎪ L̂𝓏 = -ιħ⎛cos²ϕ ∂͟ + sin²ϕ ∂͟ ⎞ +⎩ ⎝ ∂ϕ ∂ϕ⎠ + + +⎧ L̂𝓍 = ιħ⎛ sinϕ ∂͟ + cotθ cosϕ ∂͟ ⎞ +⎪ ⎝ ∂θ ∂ϕ⎠ +⎪ +⎨ L̂𝓎 = ιħ⎛-cosϕ ∂͟ + cotθ sinϕ ∂͟ ⎞ +⎪ ⎝ ∂θ ∂ϕ ⎠ +⎪ +⎪ L̂𝓏 = -ιħ∂͟ +⎩ ∂ϕ + +The spherical representation is the following set, which perfectly matches the obtained result. ⎧ L̂𝓍 = ιħ (sinϕ ∂͟_ + cosϕ cotθ ∂͟_ ) -⎪ ∂θ ∂ϕ +⎪ ∂θ ∂ϕ ⎪ ⎨ L̂𝓎 = ιħ (-cosϕ ∂͟_ + sinϕ cotθ ∂͟_ ) -⎪ ∂θ ∂ϕ +⎪ ∂θ ∂ϕ ⎪ ⎪ L̂𝓏 = -ιħ ∂͟_ -⎩ ∂ϕ - - -My set DOES NOT match this. I must be going about this the wrong way. I have to give it more thought. 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x -2151 1926 1831 2137 x -1512 2349 1161 2484 x -810 2619 396 2700 x -774 2835 1012 2929 x -1251 3024 1570 3190 x -1890 3357 2074 3528 x -2259 3699 2389 3951 x -2520 4203 2520 4500 x -2520 7200 l -5400 fwd_x -end_ol - } def -/FBA { start_ol -2538 10 m -846 3375 l -702 3375 l -459 3375 459 3564 x -459 3744 702 3744 x -1737 3744 l -1980 3744 1980 3564 x -1980 3375 1737 3375 x -1269 3375 l -2745 412 l -4194 3375 l -3708 3375 l -3465 3375 3465 3564 x -3465 3744 3708 3744 x -4698 3744 l -4941 3744 4941 3564 x -4941 3375 4698 3375 x -4572 3375 l -2268 -1287 l -2853 -1287 l -3096 -1287 3096 -1467 x -3096 -1656 2853 -1656 x -756 -1656 l -513 -1656 513 -1467 x -513 -1287 756 -1287 x -1899 -1287 l -2538 10 l -5400 fwd_x -end_ol - } def -/GBA { start_ol -4236 -567 m -4473 -567 4473 -756 x -4473 -936 4234 -936 x -3492 -936 3006 -540 x -2520 -144 2520 558 x -2520 7164 l -2889 7164 l -2889 558 l -2889 27 3264 -270 x -3641 -567 4236 -567 x -5400 fwd_x -end_ol - } def -/HBA { start_ol -4257 3744 m 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1080 x -1062 1161 1152 1251 x -2439 2538 l -1152 3816 l -1062 3915 1062 3987 x -1062 4068 1116 4122 x -1170 4176 1251 4176 x -1314 4176 1413 4077 x -2700 2799 l -5400 fwd_x -end_ol - } def -/PBA { start_ol -1881 4104 m -1881 3528 l -2091 3900 2361 4064 x -2632 4230 3035 4230 x -3785 4230 4250 3651 x -4716 3074 4716 2151 x -4716 1156 4227 524 x -3739 -108 2980 -108 x -2677 -108 2452 2 x -2228 113 1899 423 x -1899 -862 l -1899 -1318 2031 -1491 x -2164 -1665 2547 -1683 x -2547 -1872 l -144 -1872 l -144 -1656 l -468 -1602 558 -1507 x -648 -1414 648 -1136 x -648 3359 l -648 3637 553 3739 x -459 3843 162 3888 x -162 4104 l -1881 4104 l -2687 3744 m -2394 3744 2146 3496 x -1899 3249 1899 3114 x -1899 927 l -2119 378 2604 378 x -3384 378 3384 2025 x -3384 3744 2687 3744 x -5040 fwd_x -end_ol - } def -/QBA { start_ol -2025 4671 m -2025 369 l -4536 369 l -4536 1785 l -4536 2025 4725 2025 x -4905 2025 4905 1785 x -4905 0 l -820 0 l -576 0 576 189 x -576 369 813 369 x -1656 369 l -1656 4671 l -813 4671 l -576 4671 576 4860 x -576 5040 813 5040 x -2867 5040 l -3105 5040 3105 4860 x -3105 4671 2867 4671 x -2025 4671 l -5400 fwd_x -end_ol - } def -/RBA { start_ol --2727 5733 m --1539 4761 l --1422 4662 -1422 4572 x --1422 4500 -1476 4446 x --1530 4392 -1602 4392 x --1665 4392 -1764 4464 x --2727 5256 l --3690 4464 l --3789 4392 -3852 4392 x --3924 4392 -3978 4446 x --4032 4500 -4032 4572 x --4032 4644 -3915 4752 x --2727 5733 l -0 fwd_x -end_ol - } def -/SBA { start_ol -3702 3141 m -3900 3141 4024 3017 x -4149 2893 4149 2691 x -4149 2367 3778 2367 x -3582 2367 3495 2475 x -3409 2583 3396 2691 x -3384 2799 3354 2799 x -3247 2799 2970 2408 x -2692 2019 2556 1741 x -2556 1425 2443 887 x -2332 349 2332 286 x -2332 90 2478 90 x -2626 90 2783 176 x -2940 264 3171 511 x -3403 759 3490 865 x -3578 970 3865 1337 x -3934 1426 3969 1471 x -4080 1399 l -4037 1346 3852 1091 x -3667 835 3593 742 x -3519 650 3332 451 x -3146 252 3020 168 x -2895 85 2719 6 x -2542 -72 2374 -72 x -1728 -72 1521 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