diff --git a/notes/.particles in a box.swp b/notes/.particles in a box.swp new file mode 100644 index 0000000..ebd369a Binary files /dev/null and b/notes/.particles in a box.swp differ diff --git a/notes/particles in a box b/notes/particles in a box new file mode 100644 index 0000000..dc22f79 --- /dev/null +++ b/notes/particles in a box @@ -0,0 +1,67 @@ +❙Ψ❭ ≐ Ψ(x) +Ψ(x) = ❬x❙Ψ❭ +𝓟(x) = │Ψ(x)│² +𝓟(x) = ⎮Ψ(x)⎮² + + ⌠ ∞ +1 = ❬Ψ❙Ψ❭ = ⎮ │Ψ(x)│² dx = 1 + ⌡-∞ + +❙Ψ❭ → Ψ(x) +❬Ψ❙ → Ψ⃰(x) + + → A(x) + ⌠b +𝓟(a) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp 2 mul fwd_x end + } { (-) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp neg 2 mul fwd_x end + } { + % Must be a 3 char sym. Load and exec + /name ps pos 3 getinterval def + papsdict begin name draw_char end + /pos 3 pos add def + } ifelse + } ifelse + } ifelse + } ifelse + } loop + end +} def +/fontdict 1 dict def +papsdict begin fontdict begin +/AAA { start_ol +4464 0 m +2736 0 l +2736 7272 l +504 7272 l +504 8784 l +6696 8784 l +6696 7272 l +4464 7272 l +4464 0 l +7200 fwd_x +end_ol + } def +/BAA { start_ol +6264 4213 m +6264 0 l +4536 0 l +4536 3952 l +4536 4653 4340 4954 x +4144 5256 3701 5256 x +3246 5256 2991 4847 x +2736 4439 2736 3714 x +2736 0 l +1008 0 l +1008 9144 l +2736 9144 l +2736 5526 l +2916 6084 3351 6390 x +3786 6696 4401 6696 x +5323 6696 5793 6071 x +6264 5448 6264 4213 x +7200 fwd_x +end_ol + } def +/CAA { start_ol +936 2336 m +936 6552 l +2664 6552 l +2664 2596 l +2664 1896 2857 1596 x +3052 1296 3504 1296 x +3955 1296 4209 1699 x +4464 2104 4464 2833 x +4464 6552 l +6192 6552 l +6192 0 l +4464 0 l +4464 1019 l +4284 464 3852 159 x +3420 -144 2810 -144 x +1882 -144 1408 481 x +936 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+end_ol + } def +/LAA { 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 +/MAA { start_ol +4104 6854 m +1857 3384 l +4104 3384 l +4104 6854 l +3929 8784 m +5760 8784 l +5760 3384 l +6696 3384 l +6696 1872 l +5760 1872 l +5760 0 l +4104 0 l +4104 1872 l +576 1872 l +576 3560 l +3929 8784 l +7200 fwd_x +end_ol + } def +/NAA { start_ol +1080 1512 m +3024 1512 l +3024 7200 l +1224 6768 l +1224 8352 l +3021 8784 l +4680 8784 l +4680 1512 l +6624 1512 l +6624 0 l +1080 0 l +1080 1512 l +7200 fwd_x +end_ol + } def +/OAA { start_ol +3810 4464 m +3251 4464 2957 4047 x +2664 3631 2664 2840 x +2664 2056 2957 1640 x +3251 1224 3810 1224 x +4368 1224 4667 1640 x +4968 2056 4968 2840 x +4968 3625 4667 4044 x +4368 4464 3810 4464 x +6048 8568 m +6048 6984 l +5591 7272 5171 7416 x +4752 7560 4363 7560 x +3438 7560 2954 6933 x +2471 6306 2448 5081 x +2722 5456 3148 5644 x +3576 5832 4155 5832 x +5337 5832 5980 5081 x +6624 4332 6624 2955 x +6624 1479 5897 667 x +5170 -144 3852 -144 x +2245 -144 1518 927 x +792 1999 792 4388 x +792 6661 1674 7794 x +2557 8928 4323 8928 x +4732 8928 5168 8838 x +5603 8748 6048 8568 x +7200 fwd_x +end_ol + } def +/PAA { start_ol +4199 -752 m +3855 -1674 3373 -2097 x +2892 -2520 2203 -2520 x +792 -2520 l +792 -1224 l +1486 -1224 l +2011 -1224 2255 -1038 x +2500 -852 2751 -176 x +2881 174 l +360 6552 l +2192 6552 l +3685 2299 l +5115 6552 l +6912 6552 l +4199 -752 l +7200 fwd_x +end_ol + } def +/QAA { start_ol +3632 5328 m +3012 5328 2658 4785 x +2304 4243 2304 3276 x +2304 2308 2658 1766 x +3012 1224 3632 1224 x +4259 1224 4613 1766 x +4968 2308 4968 3276 x +4968 4243 4613 4785 x +4259 5328 3632 5328 x +576 3276 m +576 4852 1405 5773 x +2234 6696 3632 6696 x +5037 6696 5866 5773 x +6696 4852 6696 3276 x +6696 1699 5866 777 x +5037 -144 3632 -144 x +2234 -144 1405 777 x +576 1699 576 3276 x +7200 fwd_x +end_ol + } def +/RAA { start_ol +2664 7344 m +2664 4752 l +3369 4752 l +4214 4752 4555 5042 x +4896 5332 4896 6048 x +4896 6763 4555 7053 x +4214 7344 3369 7344 x +2664 7344 l +936 8784 m +3305 8784 l +5112 8784 5904 8139 x +6696 7494 6696 6048 x +6696 4601 5904 3956 x +5112 3312 3305 3312 x +2664 3312 l +2664 0 l +936 0 l +936 8784 l +7200 fwd_x +end_ol + } def +/SAA { start_ol +4127 3096 m +3150 3096 2763 2855 x +2376 2614 2376 2030 x +2376 1595 2643 1337 x +2911 1080 3370 1080 x +4061 1080 4442 1580 x +4824 2082 4824 2981 x +4824 3096 l +4127 3096 l +6552 3751 m +6552 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4667 x +1224 4731 1274 4779 x +1325 4827 1388 4827 x +1513 4827 1774 4390 x +2035 3955 2269 3236 x +2503 2517 2503 1894 x +2503 1302 2264 595 x +2027 -111 1761 -551 x +1497 -991 1380 -991 x +1317 -991 1270 -942 x +1224 -895 1224 -831 x +4824 fwd_x +end_ol + } def +/dAA { 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 +/eAA { 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 +/fAA { start_ol +3692 5256 m +4514 5256 5080 4943 x +5647 4630 5647 4036 x +5647 3469 5176 2893 x +4705 2318 3939 1950 x +3173 1584 2376 1584 x +2304 1584 2304 1632 x +2304 1747 2516 1881 x +2730 2016 2873 2016 x +3442 2016 3876 2206 x +4312 2397 4532 2679 x +4754 2962 4861 3224 x +4968 3487 4968 3702 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1544 793 x +1969 399 2575 399 x +3181 399 3606 788 x +4032 1179 4032 1735 x +4032 2315 3614 2703 x +3197 3092 2576 3092 x +4824 fwd_x +end_ol + } def +/ABA { 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 +/BBA { start_ol +2418 -536 m +2418 -936 l +-2376 -936 l +-2376 -536 l +2418 -536 l +0 fwd_x +end_ol + } def +/CBA { 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 +/DBA { 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 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l +2389 330 l +3931 4208 l +3308 4208 l +3092 4208 3092 4376 x +3092 4536 3308 4536 x +4507 4536 l +4722 4536 4722 4376 x +4722 4208 4507 4208 x +4259 4208 l +2589 0 l +2134 0 l +4824 fwd_x +end_ol + } def +/HBA { 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 +/IBA { start_ol +4359 1040 m +4359 849 4309 686 x +4261 524 4138 356 x +4016 190 3755 94 x +3494 0 3126 0 x +2775 0 l +2775 -1584 l +2448 -1584 l +2448 0 l +2084 0 l +1575 0 1255 277 x +936 556 936 1027 x +936 2984 l +541 2984 l +288 2984 288 3155 x +288 3312 573 3312 x +1303 3312 l +1303 1047 l +1303 723 1510 525 x +1717 327 2041 327 x +2448 327 l +2448 3312 l +3232 3312 l +3551 3312 3786 3208 x +4023 3105 4138 2938 x +4255 2772 4307 2601 x +4359 2430 4359 2255 x +4359 1040 l +3231 335 m +3660 335 3846 561 x +4032 787 4032 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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 +/MBA { 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 -914 731 -866 x +680 -817 680 -753 x +680 -690 704 -658 x +4824 fwd_x +end_ol + } def +/NBA { start_ol +1008 327 m +1008 4208 l +624 4208 l +432 4208 432 4376 x +432 4536 643 4536 x +2661 4536 l +3233 4536 3615 4197 x +3999 3858 3999 3352 x +3999 2745 3321 2368 x +4287 2010 4287 1216 x +4287 715 3911 357 x +3535 0 3010 0 x +641 0 l +432 0 432 167 x +432 327 624 327 x +1008 327 l +1335 2487 m +2507 2487 l +3018 2487 3344 2733 x +3672 2981 3672 3360 x +3672 3722 3376 3965 x +3081 4208 2643 4208 x +1335 4208 l +1335 2487 l +3960 1216 m +3960 1382 3893 1533 x +3827 1684 3677 1830 x +3529 1977 3227 2068 x +2926 2160 2503 2160 x +1335 2160 l +1335 327 l +2965 327 l +3380 327 3669 585 x +3960 842 3960 1216 x +4824 fwd_x +end_ol + } def +/OBA { start_ol +2396 3420 m +3162 3420 3688 2896 x +4215 2374 4215 1617 x +4215 884 3681 370 x +3146 -144 2396 -144 x +1638 -144 1107 374 x +576 893 576 1642 x +576 2376 1107 2898 x +1638 3420 2396 3420 x +2395 3092 m +1773 3092 1337 2669 x +903 2247 903 1641 x +903 1036 1337 609 x +1773 183 2395 183 x +3010 183 3449 605 x +3888 1028 3888 1617 x +3888 2247 3457 2669 x +3026 3092 2395 3092 x +4824 fwd_x +end_ol + } def +/PBA { start_ol +3528 0 m +3528 525 l +2855 -144 2071 -144 x +1591 -144 1299 142 x +1008 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x +455 -1184 671 -1184 x +1686 -1184 l +2253 -24 l +4824 fwd_x +end_ol + } def +/TBA { start_ol +4263 680 m +4263 609 4127 478 x +3991 347 3771 204 x +3550 61 3189 -40 x +2829 -144 2452 -144 x +1666 -144 1156 349 x +648 844 648 1609 x +648 2399 1168 2909 x +1689 3420 2491 3420 x +3238 3420 3744 2981 x +3744 3085 l +3744 3312 3911 3312 x +4071 3312 4071 3088 x +4071 2360 l +4071 2144 3911 2144 x +3767 2144 3744 2335 x +3720 2653 3346 2873 x +2973 3092 2468 3092 x +1802 3092 1388 2682 x +975 2271 975 1617 x +975 988 1392 586 x +1810 183 2468 183 x +3342 183 3967 752 x +4047 831 4110 831 x +4174 831 4218 787 x +4263 743 4263 680 x +4824 fwd_x +end_ol + } def +/UBA { start_ol +2559 3312 m +2559 327 l +3784 327 l +3999 327 3999 167 x +3999 0 3783 0 x +1000 0 l +792 0 792 167 x +792 327 1000 327 x +2232 327 l +2232 2984 l +1353 2984 l +1152 2984 1152 3152 x +1152 3312 1356 3312 x +2559 3312 l +2543 5007 m +2543 4176 l +2088 4176 l +2088 5007 l +2543 5007 l +4824 fwd_x +end_ol + } def +/VBA { 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This also means + + ❬Ψ(t=0)❙ = A⃰(❬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⎞ + ⎜0⎟ ⎜1⎟ ⎜0⎟ + ⎝0⎠, ⎝0⎠, ⎝1⎠. + +So, + ❙Ψ(t=0)❭ ≐ ⎛ A ⎞ + ⎜-A ⎟ + ⎝ιA ⎠. + +(𝐚) Multiplying the state vector by its magnitude normalizes it. + + ❙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 new file mode 100644 index 0000000..70ed598 --- /dev/null +++ b/solutions/chap5/prob2.ps @@ -0,0 +1,2317 @@ +%!PS-Adobe-3.0 +%%Title: hello +%%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 +/eBA { 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 790 x +2119 939 2351 939 x +4824 fwd_x +end_ol + } def +/fBA { 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 +/gBA { start_ol +4263 680 m +4263 609 4127 478 x +3991 347 3771 204 x +3550 61 3189 -40 x +2829 -144 2452 -144 x +1666 -144 1156 349 x +648 844 648 1609 x +648 2399 1168 2909 x +1689 3420 2491 3420 x +3238 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/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 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papsdict begin xp 2 mul fwd_x end + } { (-) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp neg 2 mul fwd_x end + } { + % Must be a 3 char sym. Load and exec + /name ps pos 3 getinterval def + papsdict begin name draw_char end + /pos 3 pos add def + } ifelse + } ifelse + } ifelse + } ifelse + } loop + end +} def +/fontdict 1 dict def +papsdict begin fontdict begin +/AAA { start_ol +4464 0 m +2736 0 l +2736 7272 l +504 7272 l +504 8784 l +6696 8784 l +6696 7272 l +4464 7272 l +4464 0 l +7200 fwd_x +end_ol + } def +/BAA { start_ol +6264 4213 m +6264 0 l +4536 0 l +4536 3952 l +4536 4653 4340 4954 x +4144 5256 3701 5256 x +3246 5256 2991 4847 x +2736 4439 2736 3714 x +2736 0 l +1008 0 l +1008 9144 l +2736 9144 l +2736 5526 l +2916 6084 3351 6390 x +3786 6696 4401 6696 x +5323 6696 5793 6071 x +6264 5448 6264 4213 x +7200 fwd_x +end_ol + } def +/CAA { start_ol +936 2336 m +936 6552 l +2664 6552 l +2664 2596 l +2664 1896 2857 1596 x +3052 1296 3504 1296 x +3955 1296 4209 1699 x +4464 2104 4464 2833 x +4464 6552 l +6192 6552 l +6192 0 l +4464 0 l +4464 1019 l +4284 464 3852 159 x +3420 -144 2810 -144 x +1882 -144 1408 481 x +936 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def +/RBA { 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 +/SBA { start_ol +2055 -1598 m +1728 -1598 l +1728 6393 l +2055 6393 l +2055 -1598 l +4824 fwd_x +end_ol + } def +/TBA { 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 +/UBA { 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 -914 731 -866 x +680 -817 680 -753 x +680 -690 704 -658 x +4824 fwd_x +end_ol + } def +/VBA { 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 790 x +2119 939 2351 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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 +Because each pair of row and column in this matrix has alternating 0s and ±1s, every multiplication operation will return 0. The second expression is therefore equivalent to the zero matrix 𝟘. \ No newline at end of file diff --git a/solutions/exam1/prob4 b/solutions/exam1/prob4 index 7f4a8e9..f02c19c 100644 --- a/solutions/exam1/prob4 +++ b/solutions/exam1/prob4 @@ -7,13 +7,13 @@  ≐ B̂ ≐ B exhibits a degenerate spectrum when it has repeated eigenvalues. 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 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. +(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 @@ -28,10 +28,10 @@ For the given operators, then, the commutator is which reduces to ⎛ ab 0 0 ⎞ ⎛ ab 0 0 ⎞ - ⎜ 0 0 ιab ⎟ - ⎜ 0 0 ιab ⎟ = 0. + ⎜ 0 0 ιab ⎟ - ⎜ 0 0 ιab ⎟ = 𝟘. ⎝ 0 -ιab 0 ⎠ ⎝ 0 -ιab 0 ⎠ -(b) Therefore, these operators commute. +(𝗯) Therefore, these operators commute. Since the operators commute, they share a set of common eigenstates. The eigenstates of  are apparent from inspection: @@ -48,14 +48,28 @@ For B̂, the eigenvalues are already known (λ = b,b,-b.), and using the eigenva ⎜ 0 0 -ιb ⎟ ⎜ β ⎟ = ⎜ -ι b γ ⎟ = b ⎜ β ⎟ ⎝ 0 ιb 0 ⎠ ⎝ γ ⎠ ⎝ ι b β ⎠ ⎝ γ ⎠ -reveals -ι γ = β, which, combined with the normalization condition, will allow the determination of two eigenstates of the B̂. One eigenstate is obvious from inspection, +dicates two possible eigenstates (for the eigenvalue b). One eigenstate is obvious from inspection: |b₁〉 ≐ ⎛1⎞ ⎜0⎟ ⎝0⎠. -If β = 1 is chosen, then γ = -ι, and if β = ι, then γ = -1. The two additional eigenstates of B̂ are therefore, after normalizing, +The eigenvalue equation also reveals -ι γ = β. If β = 1 is chosen, then γ = -ι, revealing a second eigenstate, after normalizing: - |b₂〉 ≐ 1 ⎛ 0 ⎞ |b₃〉 ≐ 1 ⎛ 0 ⎞ - √2 ⎜ 1 ⎟ √2 ⎜ ι ⎟ - ⎝-ι ⎠ and ⎝ -1 ⎠. + |b₂〉 ≐ 1 ⎛ 0 ⎞ + √2 ⎜ 1 ⎟ + ⎝-ι ⎠. + +Similarly, when the eigenvalue -b is used, the eigenvalue equation reveals ι γ = β. So, if β = ι, γ = 1. The third eigenstate is therefore, after normalizing, + + |b₃〉 ≐ 1 ⎛0⎞ + √2 ⎜ι⎟ + ⎝1⎠ + +The complete set of eigenstates of the operator B̂ is + + |b₁〉 ≐ ⎛1⎞ |b₂〉 ≐ 1 ⎛ 0 ⎞ |b₃〉 ≐ 1 ⎛0⎞ + ⎜0⎟ √2 ⎜ 1 ⎟ √2 ⎜ι⎟ + ⎝0⎠, ⎝-ι ⎠, and ⎝1⎠. + +(𝗰) These eigenstates can be expressed as linear combinations of the eigenstates of Â, so they are shared eigenstates between these operators, and so this basis is simultaneously a basis of  and B̂. diff --git a/solutions/exam1/prob4.ps b/solutions/exam1/prob4.ps index f28adda..1fedb27 100644 --- a/solutions/exam1/prob4.ps +++ b/solutions/exam1/prob4.ps @@ -1,5 +1,5 @@ %!PS-Adobe-3.0 -%%Title: Otho Ulrich: Exam 1 #4 +%%Title: yo %%Creator: paps version 0.6.7 by Dov Grobgeld %%Pages: (atend) %%BoundingBox: 0 0 595 841 @@ -178,24 +178,76 @@ end /fontdict 1 dict def papsdict begin fontdict begin /AAA { start_ol -0 8784 m -1522 8784 l -2153 2326 l -2913 6480 l -4358 6480 l -5243 2326 l -5738 8784 l -7272 8784 l -6257 0 l -4636 0 l -3632 4593 l -2695 0 l -1085 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start_ol 6480 360 m 5883 108 5263 -18 x 4644 -144 3954 -144 x @@ -221,49 +273,7 @@ end_ol 7200 fwd_x end_ol } def -/CAA { start_ol -4608 5598 m -4608 9144 l -6336 9144 l -6336 0 l -4608 0 l -4608 1019 l -4332 441 3895 148 x -3460 -144 2868 -144 x -1744 -144 1123 751 x -504 1647 504 3276 x -504 4927 1132 5811 x -1762 6696 2927 6696 x -3454 6696 3873 6421 x -4291 6147 4608 5598 x -2232 3276 m -2232 2349 2545 1822 x -2860 1296 3414 1296 x -3967 1296 4287 1822 x -4608 2349 4608 3276 x -4608 4203 4287 4729 x -3967 5256 3414 5256 x -2860 5256 2545 4729 x -2232 4203 2232 3276 x -7200 fwd_x -end_ol - } def -/EAA { start_ol -6552 7272 m -2808 7272 l -2808 5328 l -6192 5328 l -6192 3816 l -2808 3816 l -2808 0 l -1080 0 l -1080 8784 l -6552 8784 l -6552 7272 l -7200 fwd_x -end_ol - } def -/FAA { start_ol +/GAA { start_ol 4968 3276 m 4968 4203 4653 4729 x 4339 5256 3785 5256 x @@ -290,7 +300,7 @@ end_ol 7200 fwd_x end_ol } def -/GAA { start_ol +/HAA { start_ol 2543 1512 m 6203 1512 l 6203 0 l @@ 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def +/LAA { start_ol 2664 6264 m 4608 6264 l 4608 4104 l @@ -391,82 +393,9 @@ end_ol 2664 0 l 2664 2160 l 7200 fwd_x -end_ol - } def -/LAA { start_ol -3240 3960 m -2304 3960 l -2304 5472 l -3240 5472 l -3891 5472 4249 5728 x -4608 5985 4608 6450 x -4608 6940 4249 7213 x -3891 7488 3240 7488 x -2745 7488 2179 7358 x -1614 7230 1008 6984 x -1008 8568 l -1613 8742 2200 8834 x -2787 8928 3327 8928 x -4713 8928 5488 8327 x -6264 7726 6264 6670 x -6264 5893 5821 5401 x -5380 4908 4560 4756 x -5496 4592 5988 4007 x -6480 3422 6480 2470 x -6480 1195 5658 525 x -4837 -144 3272 -144 x -2604 -144 1957 -32 x -1310 77 720 288 x -720 1872 l -1278 1625 1923 1496 x -2570 1368 3281 1368 x -3993 1368 4408 1690 x -4824 2014 4824 2561 x -4824 3232 4410 3595 x -3996 3960 3240 3960 x -7200 fwd_x end_ol } def /MAA { start_ol -3668 3960 m -3090 3960 2732 3585 x -2376 3211 2376 2595 x -2376 1978 2732 1600 x -3090 1224 3668 1224 x -4248 1224 4608 1606 x -4968 1990 4968 2595 x -4968 3211 4610 3585 x -4253 3960 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-/NBA { start_ol -842 6836 m -813 6877 813 6957 x -813 7036 877 7096 x -942 7157 1012 7157 x -1062 7157 1111 7126 x -1161 7096 1191 7057 x -1220 7026 l -2592 5105 2592 2472 x -2592 -2030 l -2181 -2030 l -2181 2643 l -2181 5005 842 6836 x -5976 fwd_x -end_ol - } def -/OBA { start_ol -3384 8006 m -3794 8006 l -3794 -2001 l -3384 -2001 l -3384 8006 l -5976 fwd_x -end_ol - } def -/PBA { start_ol -5014 2592 m -991 2592 l -721 2592 721 2802 x -721 3002 991 3002 x -5014 3002 l -5284 3002 5284 2802 x -5284 2592 5014 2592 x -5976 fwd_x -end_ol - } def -/QBA { start_ol -2642 -2001 m -2232 -2001 l -2232 8006 l -2642 8006 l -2642 -2001 l -5976 fwd_x -end_ol - } def -/RBA { start_ol -4946 580 m -4946 493 4799 331 x -4653 169 4335 12 x -4018 -144 3657 -144 x -3168 -144 2880 208 x -2592 560 2592 1126 x -2592 3765 l -1993 3765 l -1728 3765 1728 3970 x -1728 4176 1994 4176 x -3002 4176 l -3002 1131 l -3002 266 3664 266 x -3916 266 4115 344 x -4314 421 4411 518 x -4509 615 4601 693 x -4693 770 4752 770 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x -1365 943 1838 605 x -2311 266 2967 266 x -3672 266 4140 630 x -4608 994 4608 1542 x -5976 fwd_x -end_ol - } def -/hBA { start_ol -2822 -6 m -940 3765 l -780 3765 l -510 3765 510 3975 x -510 4176 780 4176 x -1931 4176 l -2201 4176 2201 3975 x -2201 3765 1931 3765 x -1410 3765 l -3052 444 l -4664 3765 l -4123 3765 l -3853 3765 3853 3975 x -3853 4176 4123 4176 x -5224 4176 l -5494 4176 5494 3975 x -5494 3765 5224 3765 x -5083 3765 l -2522 -1461 l -3172 -1461 l -3442 -1461 3442 -1661 x -3442 -1872 3172 -1872 x -840 -1872 l -570 -1872 570 -1661 x -570 -1461 840 -1461 x -2111 -1461 l -2822 -6 l -5976 fwd_x -end_ol - } def -/iBA { start_ol -3218 5616 m -3218 -813 l -4176 -813 l -4442 -813 4442 -1013 x -4442 -1224 4174 -1224 x -2808 -1224 l -2808 6026 l -4174 6026 l -4442 6026 4442 5826 x -4442 5616 4176 5616 x -3218 5616 l -5976 fwd_x -end_ol - } def -/jBA { start_ol -3015 7589 m -4335 6508 l -4466 6397 4466 6297 x -4466 6217 4405 6157 x -4345 6097 4266 6097 x -4196 6097 4086 6177 x -3015 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