From 4c4d4d0ead84935e8424050c1a0033d8c5f6481f Mon Sep 17 00:00:00 2001 From: othocaes Date: Wed, 24 Feb 2016 15:38:17 -0500 Subject: [PATCH] still working on exam 1 --- solutions/exam1/prob1 | 34 +- solutions/exam1/prob1.ps | 472 ++++--- solutions/exam1/prob3 | 87 +- solutions/exam1/prob3.ps | 2786 ++++++++++++++++++++++++++++++++++++++ solutions/exam1/prob4 | 47 + solutions/exam1/prob4.ps | 2631 +++++++++++++++++++++++++++++++++++ 6 files changed, 5847 insertions(+), 210 deletions(-) create mode 100644 solutions/exam1/prob3.ps create mode 100644 solutions/exam1/prob4 create mode 100644 solutions/exam1/prob4.ps diff --git a/solutions/exam1/prob1 b/solutions/exam1/prob1 index e3e4d8f..0a8f8fc 100644 --- a/solutions/exam1/prob1 +++ b/solutions/exam1/prob1 @@ -4,36 +4,38 @@ The spin eigenstates and eigenvalues are known from experiment for a spin-1 syst S𝓏 ≐ - ⎛ 1 0 0 ⎞ - ħ ⎜ 0 0 0 ⎟ and + ħ ⎛ 1 0 0 ⎞ + ⎜ 0 0 0 ⎟ and ⎝ 0 0 -1 ⎠ S𝓍 ≐ - ⎛ 0 1 0 ⎞ - ħ ⎜ 1 0 1 ⎟. - ⎝ 0 1 0 ⎠ + ͟ħ͟ ⎛ 0 1 0 ⎞ + √2 ⎜ 1 0 1 ⎟. + ⎝ 0 1 0 ⎠ Using the matrix representations, the expressions can be evaluated. For the spin-z operator, the expression S𝓏 (S𝓏 + ħ)(S𝓏 - ħ) ≐ - ⎛ 1 0 0 ⎞ ⎧ ⎛ 1 0 0 ⎞ ⎛ 1 0 0 ⎞ ⎫ ⎧ ⎛ 1 0 0 ⎞ ⎛ 1 0 0 ⎞ ⎫ - ħ ⎜ 0 0 0 ⎟ ⎪ ħ ⎜ 0 0 0 ⎟ + ħ ⎜ 0 1 0 ⎟ ⎪ ⎪ ħ ⎜ 0 0 0 ⎟ - ħ ⎜ 0 1 0 ⎟ ⎪, + ħ ⎛ 1 0 0 ⎞ ⎧ ħ ⎛ 1 0 0 ⎞ ħ ⎛ 1 0 0 ⎞ ⎫ ⎧ ħ ⎛ 1 0 0 ⎞ ħ ⎛ 1 0 0 ⎞ ⎫ + ⎜ 0 0 0 ⎟ ⎪ ⎜ 0 0 0 ⎟ + ⎜ 0 1 0 ⎟ ⎪ ⎪ ⎜ 0 0 0 ⎟ - ⎜ 0 1 0 ⎟ ⎪, ⎝ 0 0 -1 ⎠ ⎩ ⎝ 0 0 -1 ⎠ ⎝ 0 0 1 ⎠ ⎭ ⎩ ⎝ 0 0 -1 ⎠ ⎝ 0 0 1 ⎠ ⎭ which simplifies to the matrix multiplication operation, where 𝟘 represents the 0 matrix, - ⎛ 1 0 0 ⎞ ⎛ 2 0 0 ⎞ ⎛ 0 0 0 ⎞ - ħ ⎜ 0 0 0 ⎟ ħ ⎜ 0 1 0 ⎟ ħ ⎜ 0 -1 0 ⎟ = 𝟘. + ħ ⎛ 1 0 0 ⎞ ħ ⎛ 2 0 0 ⎞ ħ ⎛ 0 0 0 ⎞ + ⎜ 0 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 -1 0 ⎟ = 𝟘. ⎝ 0 0 -1 ⎠ ⎝ 0 0 0 ⎠ ⎝ 0 0 0 ⎠ The multiplication operation apparently returns 𝟘 because the third factor will nullify any terms besides center terms, and the first factor will nullify any center terms. Similarly, S𝓍 (S𝓍 + ħ)(S𝓍 - ħ) ≐ - ⎛ 0 1 0 ⎞ ⎛ 1 1 0 ⎞ ⎛ -1 1 0 ⎞ ⎛ 1 1 1 ⎞ ⎛ -1 1 0 ⎞ ⎛ 0 1 0 ⎞ - ħ³ ⎜ 1 0 1 ⎟ ⎜ 1 1 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 2 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 0 1 ⎟. - ⎝ 0 1 0 ⎠ ⎝ 0 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 1 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 0 1 0 ⎠ + ͟ħ͟ ⎛ 0 1 0 ⎞ ͟ħ͟ ⎛ √2 1 0 ⎞ ͟ħ͟ ⎛ -√2 1 0 ⎞ + √2 ⎜ 1 0 1 ⎟ √2 ⎜ 1 √2 1 ⎟ √2 ⎜ 1 -√2 1 ⎟. + ⎝ 0 1 0 ⎠ ⎝ 0 1 √2 ⎠ ⎝ 0 1 -√2 ⎠ -This expression results in the non-zero matrix +Performing the multiplication operation on the last two matrices returns the expression - ⎛ 0 1 0 ⎞ - ħ³ ⎜ 1 0 1 ⎟. - ⎝ 0 1 0 ⎠ \ No newline at end of file + ͟ħ͟³͟ ⎛ 0 1 0 ⎞ ⎛ -1 0 1 ⎞ + 2√2 ⎜ 1 0 1 ⎟ ⎜ 0 0 0 ⎟ = 𝟘. + ⎝ 0 1 0 ⎠ ⎝ 1 0 -1 ⎠ + +It is quite obvious that this operation returns 𝟘 since there are no components that will not match with a 0 throughout the multiplication of these matrices. Therefore, the second expression is also equivalent to the zero matrix 𝟘. \ No newline at end of file diff --git a/solutions/exam1/prob1.ps b/solutions/exam1/prob1.ps index 7ad8ea1..e396bf7 100644 --- a/solutions/exam1/prob1.ps +++ b/solutions/exam1/prob1.ps @@ -1,5 +1,5 @@ %!PS-Adobe-3.0 -%%Title: Otho Ulrich: Exam 1, #1 +%%Title: Otho Ulrich: Exam 1 #1 %%Creator: paps version 0.6.7 by Dov Grobgeld %%Pages: (atend) %%BoundingBox: 0 0 595 841 @@ -353,37 +353,6 @@ end_ol end_ol } def /JAA { start_ol -2880 4388 m -2880 4695 3088 4903 x -3296 5112 3596 5112 x -3903 5112 4111 4903 x -4320 4695 4320 4388 x -4320 4088 4111 3880 x -3903 3672 3596 3672 x -3296 3672 3088 3876 x -2880 4082 2880 4388 x -3596 7488 m -2999 7488 2723 6750 x -2448 6012 2448 4388 x -2448 2772 2723 2034 x -2999 1296 3596 1296 x -4200 1296 4476 2034 x -4752 2772 4752 4388 x -4752 6012 4476 6750 x -4200 7488 3596 7488 x -720 4388 m -720 6661 1437 7794 x -2155 8928 3596 8928 x -5044 8928 5762 7797 x -6480 6666 6480 4388 x -6480 2116 5762 985 x -5044 -144 3596 -144 x -2155 -144 1437 988 x -720 2122 720 4388 x -7200 fwd_x -end_ol - } def -/KAA { start_ol 2664 6264 m 4608 6264 l 4608 4104 l @@ -395,6 +364,37 @@ end_ol 2664 0 l 2664 2160 l 7200 fwd_x +end_ol + } def +/KAA { start_ol +1224 216 m +1224 1800 l +1680 1505 2100 1364 x +2520 1224 2908 1224 x +3828 1224 4311 1847 x +4794 2470 4824 3710 x +4549 3331 4122 3141 x +3695 2952 3110 2952 x +1935 2952 1291 3703 x +648 4456 648 5836 x +648 7311 1371 8119 x +2095 8928 3420 8928 x +5026 8928 5753 7857 x +6480 6786 6480 4397 x +6480 2127 5596 991 x +4714 -144 2948 -144 x +2539 -144 2103 -51 x +1668 39 1224 216 x +3465 4392 m +4018 4392 4313 4808 x +4608 5224 4608 6015 x +4608 6799 4313 7215 x +4018 7632 3465 7632 x +2904 7632 2604 7215 x +2304 6799 2304 6015 x +2304 5230 2604 4810 x +2904 4392 3465 4392 x +7200 fwd_x end_ol } def /LAA { start_ol @@ -432,33 +432,33 @@ end_ol end_ol } def /MAA { start_ol -1224 216 m -1224 1800 l -1680 1505 2100 1364 x -2520 1224 2908 1224 x -3828 1224 4311 1847 x -4794 2470 4824 3710 x -4549 3331 4122 3141 x -3695 2952 3110 2952 x -1935 2952 1291 3703 x -648 4456 648 5836 x -648 7311 1371 8119 x -2095 8928 3420 8928 x -5026 8928 5753 7857 x -6480 6786 6480 4397 x -6480 2127 5596 991 x -4714 -144 2948 -144 x -2539 -144 2103 -51 x -1668 39 1224 216 x -3465 4392 m -4018 4392 4313 4808 x -4608 5224 4608 6015 x -4608 6799 4313 7215 x -4018 7632 3465 7632 x -2904 7632 2604 7215 x -2304 6799 2304 6015 x -2304 5230 2604 4810 x -2904 4392 3465 4392 x +2880 4388 m +2880 4695 3088 4903 x +3296 5112 3596 5112 x +3903 5112 4111 4903 x +4320 4695 4320 4388 x +4320 4088 4111 3880 x +3903 3672 3596 3672 x +3296 3672 3088 3876 x +2880 4082 2880 4388 x +3596 7488 m +2999 7488 2723 6750 x +2448 6012 2448 4388 x +2448 2772 2723 2034 x +2999 1296 3596 1296 x +4200 1296 4476 2034 x +4752 2772 4752 4388 x +4752 6012 4476 6750 x +4200 7488 3596 7488 x +720 4388 m +720 6661 1437 7794 x +2155 8928 3596 8928 x +5044 8928 5762 7797 x +6480 6666 6480 4388 x +6480 2116 5762 985 x +5044 -144 3596 -144 x +2155 -144 1437 988 x +720 2122 720 4388 x 7200 fwd_x end_ol } def @@ -782,17 +782,6 @@ end_ol end_ol } def /bAA { start_ol -2736 2160 m -4680 2160 l -4680 578 l -3456 -1656 l -2160 -1656 l -2736 578 l -2736 2160 l -7200 fwd_x -end_ol - } def -/cAA { start_ol 4136 8640 m 3579 6408 l 4771 6408 l @@ -830,7 +819,7 @@ end_ol 7200 fwd_x end_ol } def -/dAA { start_ol +/cAA { start_ol 2664 7344 m 2664 4752 l 3369 4752 l @@ -852,7 +841,7 @@ end_ol 7200 fwd_x end_ol } def -/eAA { start_ol +/dAA { start_ol 4680 3351 m 4680 4214 4351 4735 x 4023 5256 3486 5256 x @@ -887,7 +876,7 @@ end_ol 7200 fwd_x end_ol } def -/fAA { start_ol +/eAA { start_ol 3218 5205 m 3218 410 l 4596 410 l @@ -908,7 +897,7 @@ end_ol 5976 fwd_x end_ol } def -/gAA { start_ol +/fAA { start_ol 5171 5544 m 5151 5544 4924 5562 x 4698 5580 4377 5598 x @@ -942,7 +931,7 @@ end_ol 5976 fwd_x end_ol } def -/iAA { start_ol +/hAA { start_ol 4608 1542 m 4608 1960 4344 2219 x 4082 2479 3693 2563 x @@ -985,7 +974,7 @@ end_ol 5976 fwd_x end_ol } def -/jAA { start_ol +/iAA { start_ol 2724 2941 m 2724 2710 2459 2378 x 2194 2046 2194 2036 x @@ -1037,7 +1026,7 @@ end_ol 4320 fwd_x end_ol } def -/lAA { start_ol +/kAA { start_ol 1249 3812 m 1249 4023 1926 4171 x 2603 4320 2981 4320 x @@ -1074,7 +1063,7 @@ end_ol 5976 fwd_x end_ol } def -/mAA { start_ol +/lAA { start_ol 3207 3909 m 3029 3909 2866 3873 x 2703 3838 2560 3763 x @@ -1115,7 +1104,7 @@ end_ol 5976 fwd_x end_ol } def -/nAA { start_ol +/mAA { start_ol 5018 6026 m 5018 410 l 5546 410 l @@ -1146,7 +1135,7 @@ end_ol 5976 fwd_x end_ol } def -/oAA { start_ol +/nAA { start_ol 4097 3450 m 4317 3450 4456 3313 x 4594 3175 4594 2953 x @@ -1194,7 +1183,7 @@ end_ol 4392 fwd_x end_ol } def -/pAA { start_ol +/oAA { start_ol 5221 3444 m 5170 3444 4910 3658 x 4650 3873 4399 3873 x @@ -1223,7 +1212,7 @@ end_ol 5976 fwd_x end_ol } def -/qAA { start_ol +/pAA { start_ol 5204 2016 m 1040 2016 l 1150 1221 1701 743 x @@ -1250,7 +1239,7 @@ end_ol 5976 fwd_x end_ol } def -/rAA { start_ol +/qAA { start_ol 4536 1179 m 4536 1545 4189 1743 x 3843 1942 3362 2017 x @@ -1292,7 +1281,7 @@ end_ol 5976 fwd_x end_ol } def -/sAA { start_ol +/rAA { start_ol 1418 4176 m 1418 3351 l 1825 3875 2242 4097 x @@ -1327,7 +1316,7 @@ end_ol 5976 fwd_x end_ol } def -/tAA { start_ol +/sAA { start_ol 3218 4176 m 3218 410 l 4811 410 l @@ -1350,7 +1339,7 @@ end_ol 5976 fwd_x end_ol } def -/uAA { start_ol +/tAA { start_ol 5014 2592 m 991 2592 l 721 2592 721 2802 x @@ -1361,7 +1350,7 @@ end_ol 5976 fwd_x end_ol } def -/vAA { start_ol +/uAA { start_ol 3218 6142 m 3218 410 l 4605 410 l @@ -1382,7 +1371,7 @@ end_ol 5976 fwd_x end_ol } def -/wAA { start_ol +/vAA { start_ol 3013 4320 m 3979 4320 4642 3665 x 5306 3011 5306 2063 x @@ -1404,7 +1393,7 @@ end_ol 5976 fwd_x end_ol } def -/xAA { start_ol +/wAA { start_ol 1850 4176 m 4030 4176 l 4298 4176 4298 3975 x @@ -1434,7 +1423,7 @@ end_ol 5976 fwd_x end_ol } def -/yAA { start_ol +/xAA { start_ol 4392 2886 m 4392 3337 4046 3623 x 3702 3909 3145 3909 x @@ -1471,7 +1460,7 @@ end_ol 5976 fwd_x end_ol } def -/zAA { start_ol +/yAA { start_ol 4658 4176 m 4658 3816 l 1694 410 l @@ -1492,7 +1481,7 @@ end_ol 5976 fwd_x end_ol } def -/ABA { start_ol +/zAA { start_ol 1418 6026 m 1418 3288 l 2161 4320 3250 4320 x @@ -1523,7 +1512,7 @@ end_ol 5976 fwd_x end_ol } def -/BBA { start_ol +/ABA { start_ol 2089 1462 m 3420 1462 l 1968 -1190 l @@ -1535,7 +1524,7 @@ end_ol 5976 fwd_x end_ol } def -/CBA { start_ol +/BBA { start_ol 4413 0 m 3913 0 l 3006 2594 l @@ -1565,7 +1554,7 @@ end_ol 5976 fwd_x end_ol } def -/DBA { start_ol +/CBA { start_ol 4392 0 m 4392 661 l 3547 -144 2560 -144 x @@ -1593,7 +1582,7 @@ end_ol 5976 fwd_x end_ol } def -/EBA { start_ol +/DBA { start_ol 3218 6026 m 3218 410 l 4811 410 l @@ -1611,7 +1600,7 @@ end_ol 5976 fwd_x end_ol } def -/FBA { start_ol +/EBA { start_ol 2952 2432 m 2952 3172 3252 4057 x 3553 4944 3889 5494 x @@ -1632,7 +1621,7 @@ end_ol 5976 fwd_x end_ol } def -/GBA { start_ol +/FBA { start_ol 3218 2592 m 3218 623 l 3218 360 3018 360 x @@ -1653,7 +1642,7 @@ end_ol 5976 fwd_x end_ol } def -/HBA { start_ol +/GBA { start_ol 4392 2886 m 4392 3337 4046 3623 x 3702 3909 3145 3909 x @@ -1700,7 +1689,7 @@ end_ol 5976 fwd_x end_ol } def -/IBA { start_ol +/HBA { start_ol 1512 -1040 m 1512 -930 1668 -634 x 1824 -339 2016 14 x @@ -1721,7 +1710,7 @@ end_ol 5976 fwd_x end_ol } def -/KBA { start_ol +/JBA { start_ol 3362 0 m 2661 0 l 981 3765 l @@ -1746,7 +1735,7 @@ end_ol 5976 fwd_x end_ol } def -/LBA { start_ol +/KBA { start_ol 1368 5278 m 1465 5308 1769 5454 x 2074 5600 2383 5680 x @@ -1782,7 +1771,7 @@ end_ol 5976 fwd_x end_ol } def -/MBA { start_ol +/LBA { start_ol 3218 410 m 4249 410 l 4514 410 4514 210 x @@ -1808,7 +1797,7 @@ end_ol 5976 fwd_x end_ol } def -/NBA { start_ol +/MBA { start_ol 4392 3341 m 4392 4176 l 5328 4176 l @@ -1843,7 +1832,7 @@ end_ol 5976 fwd_x end_ol } def -/OBA { start_ol +/NBA { start_ol 1850 1802 m 1850 0 l 914 0 l @@ -1878,7 +1867,7 @@ end_ol 5976 fwd_x end_ol } def -/PBA { start_ol +/OBA { start_ol 1130 4176 m 1130 3655 l 1377 4017 1613 4168 x @@ -1921,7 +1910,7 @@ end_ol 5976 fwd_x end_ol } def -/QBA { start_ol +/PBA { start_ol 3292 2192 m 5134 410 l 5224 410 l @@ -1962,7 +1951,7 @@ end_ol 5976 fwd_x end_ol } def -/RBA { start_ol +/QBA { start_ol 2822 -6 m 940 3765 l 780 3765 l @@ -1993,7 +1982,7 @@ end_ol 5976 fwd_x end_ol } def -/SBA { start_ol +/RBA { start_ol 2922 1166 m 3022 1166 l 3313 1166 3513 981 x @@ -2019,7 +2008,7 @@ end_ol 5976 fwd_x end_ol } def -/TBA { start_ol +/SBA { start_ol 3020 5105 m 3231 5105 3376 4954 x 3521 4804 3521 4594 x @@ -2046,7 +2035,7 @@ end_ol 5976 fwd_x end_ol } def -/UBA { start_ol +/TBA { start_ol 5133 6865 m 3794 5034 3794 2671 x 3794 -2001 l @@ -2060,7 +2049,7 @@ end_ol 5976 fwd_x end_ol } def -/VBA { start_ol +/UBA { start_ol 4874 3506 m 4874 2509 l 4874 1332 4356 594 x @@ -2086,7 +2075,7 @@ end_ol 5976 fwd_x end_ol } def -/WBA { start_ol +/VBA { start_ol 842 6836 m 813 6877 813 6957 x 813 7036 877 7096 x @@ -2102,7 +2091,7 @@ end_ol 5976 fwd_x end_ol } def -/XBA { start_ol +/WBA { start_ol 3384 8006 m 3794 8006 l 3794 -2001 l @@ -2111,7 +2100,7 @@ end_ol 5976 fwd_x end_ol } def -/YBA { start_ol +/XBA { start_ol 2642 -2001 m 2232 -2001 l 2232 8006 l @@ -2120,7 +2109,7 @@ end_ol 5976 fwd_x end_ol } def -/ZBA { start_ol +/YBA { start_ol 5133 -860 m 5162 -901 5162 -981 x 5162 -1060 5097 -1120 x @@ -2136,7 +2125,7 @@ end_ol 5976 fwd_x end_ol } def -/aBA { start_ol +/ZBA { start_ol 842 -889 m 2181 941 2181 3304 x 2181 7977 l @@ -2148,9 +2137,65 @@ end_ol 813 -1090 813 -1009 x 813 -929 842 -889 x 5976 fwd_x +end_ol + } def +/aBA { start_ol +2981 -651 m +2981 -1152 l +-3024 -1152 l +-3024 -651 l +2981 -651 l +0 fwd_x end_ol } def /bBA { start_ol +1271 3146 m +1821 995 l +3822 6984 l +5784 6984 l +6005 6984 6005 6804 x +6005 6624 5784 6624 x +4113 6624 l +1791 0 l +950 2736 l +320 2736 l +100 2736 100 2946 x +100 3146 320 3146 x +1271 3146 l +5976 fwd_x +end_ol + } def +/cBA { start_ol +1008 4692 m +1008 5142 1562 5656 x +2117 6170 2900 6170 x +3639 6170 4200 5650 x +4763 5132 4763 4452 x +4763 4003 4494 3609 x +4225 3215 3303 2367 x +1202 441 l +1202 410 l +4392 410 l +4392 770 l +4392 1040 4602 1040 x +4802 1040 4802 770 x +4802 0 l +864 0 l +864 600 l +3222 2815 l +3921 3504 4136 3813 x +4351 4123 4351 4462 x +4351 4981 3915 5370 x +3478 5760 2900 5760 x +2382 5760 1960 5460 x +1540 5161 1418 4711 x +1356 4512 1202 4512 x +1131 4512 1069 4567 x +1008 4622 1008 4692 x +5976 fwd_x +end_ol + } def +/dBA { start_ol 2922 1166 m 3022 1166 l 3313 1166 3513 981 x @@ -2165,7 +2210,7 @@ end_ol 5976 fwd_x end_ol } def -/cBA { start_ol +/eBA { start_ol 5018 5205 m 5018 1854 l 5018 1009 4439 432 x @@ -2196,7 +2241,7 @@ end_ol 5976 fwd_x end_ol } def -/dBA { start_ol +/fBA { start_ol 5330 873 m 5330 785 5160 623 x 4990 462 4718 285 x @@ -2226,7 +2271,7 @@ end_ol 5976 fwd_x end_ol } def -/eBA { start_ol +/gBA { start_ol 1634 2736 m 1634 410 l 3011 410 l @@ -2259,7 +2304,7 @@ end_ol 5976 fwd_x end_ol } def -/fBA { start_ol +/hBA { start_ol 4749 6624 m 4073 6624 3646 6294 x 3218 5965 3218 5377 x @@ -2273,7 +2318,7 @@ end_ol 5976 fwd_x end_ol } def -/gBA { start_ol +/iBA { start_ol 1226 6573 m 957 6573 957 6783 x 957 6984 1227 6984 x @@ -2287,7 +2332,7 @@ end_ol 5976 fwd_x end_ol } def -/hBA { start_ol +/jBA { start_ol 3218 8006 m 3218 -2001 l 2808 -2001 l @@ -2296,7 +2341,7 @@ end_ol 5976 fwd_x end_ol } def -/iBA { start_ol +/kBA { start_ol 4749 -597 m 5018 -597 5018 -807 x 5018 -1008 4748 -1008 x @@ -2310,7 +2355,7 @@ end_ol 5976 fwd_x end_ol } def -/jBA { start_ol +/lBA { start_ol 1226 -648 m 1902 -648 2329 -318 x 2757 10 2757 598 x @@ -2324,7 +2369,7 @@ end_ol 5976 fwd_x end_ol } def -/kBA { start_ol +/mBA { start_ol 1309 6457 m 1955 7428 3181 7428 x 4407 7428 5058 6457 x @@ -2352,37 +2397,7 @@ end_ol 6336 fwd_x end_ol } def -/mBA { start_ol -1008 4692 m -1008 5142 1562 5656 x -2117 6170 2900 6170 x -3639 6170 4200 5650 x -4763 5132 4763 4452 x -4763 4003 4494 3609 x -4225 3215 3303 2367 x -1202 441 l -1202 410 l -4392 410 l -4392 770 l -4392 1040 4602 1040 x -4802 1040 4802 770 x -4802 0 l -864 0 l -864 600 l -3222 2815 l -3921 3504 4136 3813 x -4351 4123 4351 4462 x -4351 4981 3915 5370 x -3478 5760 2900 5760 x -2382 5760 1960 5460 x -1540 5161 1418 4711 x -1356 4512 1202 4512 x -1131 4512 1069 4567 x -1008 4622 1008 4692 x -5976 fwd_x -end_ol - } def -/nBA { start_ol +/oBA { start_ol 5224 3312 m 780 3312 l 510 3312 510 3522 x @@ -2400,7 +2415,38 @@ end_ol 5976 fwd_x end_ol } def -/oBA { start_ol +/pBA { start_ol +1634 2325 m +1634 410 l +3011 410 l +3290 410 3290 210 x +3290 0 3012 0 x +699 0 l +432 0 432 210 x +432 410 696 410 x +1224 410 l +1224 5205 l +696 5205 l +432 5205 432 5415 x +432 5616 703 5616 x +3117 5616 l +3921 5616 4469 5143 x +5018 4672 5018 3980 x +5018 3289 4420 2806 x +3822 2325 2962 2325 x +1634 2325 l +1634 2736 m +2994 2736 l +3654 2736 4131 3101 x +4608 3467 4608 3980 x +4608 4484 4176 4844 x +3745 5205 3146 5205 x +1634 5205 l +1634 2736 l +5976 fwd_x +end_ol + } def +/qBA { start_ol 3013 6984 m 2810 6984 2655 6925 x 2499 6867 2436 6804 x @@ -2434,6 +2480,40 @@ end_ol 3672 6666 3483 6825 x 3294 6984 3013 6984 x 5976 fwd_x +end_ol + } def +/rBA { start_ol +4608 3351 m +4608 4176 l +5544 4176 l +5810 4176 5810 3978 x +5810 3765 5546 3765 x +5018 3765 l +5018 -1461 l +5546 -1461 l +5810 -1461 5810 -1661 x +5810 -1872 5545 -1872 x +3649 -1872 l +3384 -1872 3384 -1661 x +3384 -1461 3648 -1461 x +4608 -1461 l +4608 1038 l +3942 72 2818 72 x +1902 72 1274 686 x +648 1300 648 2199 x +648 3098 1274 3709 x +1902 4320 2828 4320 x +3931 4320 4608 3351 x +2828 3909 m +2082 3909 1570 3413 x +1058 2917 1058 2199 x +1058 1482 1570 982 x +2082 482 2828 482 x +3564 482 4086 977 x +4608 1472 4608 2179 x +4608 2918 4096 3413 x +3584 3909 2828 3909 x +5976 fwd_x end_ol } def end end @@ -2441,50 +2521,60 @@ end end %%EndPrologue %%Page: 1 1 paps_bop -(+ 36000 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a/solutions/exam1/prob3 b/solutions/exam1/prob3 index 7a5a9b0..edf5f8d 100644 --- a/solutions/exam1/prob3 +++ b/solutions/exam1/prob3 @@ -1,8 +1,89 @@ |a〉 and |b〉 are eigenstates of a Hermitian operator A with eigenvalues a and b, a ≠ b. The Hamiltonian operator is - Ĥ = |a〉 δ 〈b| + |b〉 δ 〈a|, with δ a real number. + Ĥ = |a〉 δ 〈a| + |b〉 δ 〈b|, with δ a real number. -a) The eigenstates of the Hamiltonian can be determined by +a) The eigenstates of the Hamiltonian can be determined by diagonalizing the Hamiltonian operator's matrix representation. In general, -🔋 +Ĥ ≐ + ⎛ 〈a|Ĥ|a〉 〈a|Ĥ|b〉 ⎞ + ⎝ 〈b|Ĥ|a〉 〈b|Ĥ|b〉 ⎠. +Calculating the individual components: + + 〈a|Ĥ|a〉 = 〈a|(|a〉 δ 〈a| + |b〉 δ 〈b|)|a〉 = + + 〈a|Ĥ|a〉 = 〈a|a〉 δ 〈a|a〉 + 〈a|b〉 δ 〈b|a〉 = + + 〈a|Ĥ|a〉 = δ(1 + 〈a|b〉〈b|a〉), + +and because H is a hermitian operator, 〈a|b〉 = 〈b|a〉, so + + 〈a|Ĥ|a〉 = δ(1 + |〈a|b〉|²); + + 〈a|Ĥ|b〉 = 〈a|a〉 δ 〈a|b〉 + 〈a|b〉 δ 〈b|b〉 = + + 〈a|Ĥ|b〉 = δ (〈a|b〉 + 〈a|b〉) = δ 2〈a|b〉; + +because of the hermition property, + 〈b|Ĥ|a〉 = 〈a|Ĥ|b〉 = δ 2〈a|b〉; + +finally, + + 〈b|Ĥ|b〉 = 〈b|(|a〉 δ 〈a| + |b〉 δ 〈b|)|b〉 = + + 〈b|Ĥ|b〉 = 〈b|a〉 δ 〈a|b〉 + 〈b|b〉 δ 〈b|b〉 = + + 〈b|Ĥ|b〉 = δ(〈b|a〉〈a|b〉 + 1), + + 〈b|Ĥ|b〉 = δ(1 + |〈a|b〉|²). + +So, the Hamiltonian operator Ĥ ≐ + + δ ⎛ 1 + |〈a|b〉|² 2〈a|b〉 ⎞ + ⎝ 2〈a|b〉 1 + |〈a|b〉|² ⎠. + +The eigenstates, which I will call |1〉 and |2〉 can be obtained by diagonalizing the Hamiltonian matrix. The first eigenvalue equations are + + Ĥ|1〉 = E₁|1〉 and Ĥ|2〉 = E₂|2〉, with the eigenstates represented by the vector matrices, respectively, + + ⎛α₁⎞ ⎛α₂⎞ + ⎝β₁⎠ and ⎝β₂⎠. + + δ ⎛ 1 + |〈a|b〉|² 2〈a|b〉 ⎞ ⎛α₁⎞ = E₁ ⎛α₁⎞ + ⎝ 2〈a|b〉 1 + |〈a|b〉|² ⎠ ⎝β₁⎠ ⎝β₁⎠. + +This gives the equation α₁ + α₁|〈a|b〉|² + 2β₁〈a|b〉 = E₁α₁, and therefore the ratio between α₁ and β₁, + + ͟β͟₁͟ = ͟E͟₁͟ ͟-͟ ͟1͟ ͟-͟ ͟|͟〈͟a͟|͟b͟〉͟|͟²͟, or + α₁ 2〈a|b〉 + + β₁ = ͟α͟₁͟(͟E͟₁͟ ͟-͟ ͟1͟ ͟-͟ ͟|͟〈͟a͟|͟b͟〉͟|͟²͟)͟ + 2〈a|b〉 + +Using the normalization condition, the values of each constant can be obtained. Plugging the value for α₁ into the equation reveals a quadratic equation. + + + + |α₁|² + |β₁|² = 1, so + + |α₁|² + | ͟α͟₁͟(͟E͟₁͟ ͟-͟ ͟1͟ ͟-͟ ͟|͟〈͟a͟|͟b͟〉͟|͟²͟)͟ |² = 1. + | 2〈a|b〉 | + + + + + + α₁(1 - E₁) + 2β₁〈a|b〉 + α₁|〈a|b〉|² = 0 and + _________ + α₁ = ±√1 - |β₁|², so + _________ _________ + ±√̅1 - |β₁|² (1 - E₁) + 2β₁〈a|b〉 + ±√1 - |β₁|² |〈a|b〉|² = 0. + + The quadratic formula therefore says that + + + + + + + \ No newline at end of file diff --git a/solutions/exam1/prob3.ps b/solutions/exam1/prob3.ps new file mode 100644 index 0000000..bb9bd60 --- /dev/null +++ b/solutions/exam1/prob3.ps @@ -0,0 +1,2786 @@ +%!PS-Adobe-3.0 +%%Title: Otho Ulrich: Exam 1 #3 +%%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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The eigenvalues of B are obtained from its characteristic equation. + +|⎛ b-λ 0 0 ⎞| +|⎜ 0 -λ -ιb ⎟| = 0, i.e. +|⎝ 0 ιb -λ ⎠| + +(b - λ)(λ² + ι²b²) = (b - λ)(λ² - b²) = (b - λ)(b - λ)(b + λ) = 0. + +(a) The eigenvalues for this operator are therefore λ = b,b,-b. Since b appears twice, the operator exhibits a degenerate spectrum. + +To find if A and B commute, their commutator need be evaluated. They commute if the value is 0. The commutator of two operators is defined as + + [Â,B̂] = Â B̂ - B̂ Â. + +For the given operators, then, the commutator is + + ⎛ a 0 0 ⎞ ⎛ b 0 0 ⎞ ⎛ b 0 0 ⎞ ⎛ a 0 0 ⎞ + ⎜ 0 -a 0 ⎟ ⎜ 0 0 -ιb ⎟ - ⎜ 0 0 -ιb ⎟ ⎜ 0 -a 0 ⎟ + ⎝ 0 0 -a ⎠ ⎝ 0 ιb 0 ⎠ ⎝ 0 ιb 0 ⎠ ⎝ 0 0 -a ⎠, + +which reduces to + + ⎛ ab 0 0 ⎞ ⎛ ab 0 0 ⎞ + ⎜ 0 0 ιab ⎟ - ⎜ 0 0 ιab ⎟ = 0. + ⎝ 0 -ιab 0 ⎠ ⎝ 0 -ιab 0 ⎠ + +(b) Therefore, these operators commute. + +Since the operators commute, they share common eigenstates. Therefore, if the eigenstates for one operator can be determined, they are determined for both operators. Since the problem states that a new set of orthonormal kets need be determined, and the given set are the eigenstates related to the operator A, then the eigenstates of the operator B should be determined. + +The eigenvalues are already known (λ = b,b,-b.), and using the eigenvalue equations, the eigenstates can be determined. The eigenvalue equation + + ⎛ b 0 0 ⎞ ⎛ α₁ ⎞ ⎛ b α₁ ⎞ ⎛ α₁ ⎞ + ⎜ 0 0 -ιb ⎟ ⎜ β₁ ⎟ = ⎜ -ι b γ₁ ⎟ = b ⎜ β₁ ⎟ + ⎝ 0 ιb 0 ⎠ ⎝ γ₁ ⎠ ⎝ ι b β₁ ⎠ ⎝ γ₁ ⎠ reveals + + -ι γ₁ = β₁, and + ι b β₁ = γ₁ + + diff --git a/solutions/exam1/prob4.ps b/solutions/exam1/prob4.ps new file mode 100644 index 0000000..f28adda --- /dev/null +++ b/solutions/exam1/prob4.ps @@ -0,0 +1,2631 @@ +%!PS-Adobe-3.0 +%%Title: Otho Ulrich: Exam 1 #4 +%%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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