diff --git a/lecture_notes/1-27/Constant of Motion b/lecture_notes/1-27/Constant of Motion new file mode 100644 index 0000000..88b1a88 --- /dev/null +++ b/lecture_notes/1-27/Constant of Motion @@ -0,0 +1,10 @@ +Compatible Observables occur when commutator is equal to 0. + +[Â,B̂] = ÂB̂ - B̂ + +Constant of Motion +------------------ + if d/dt  = 0 and [Ĥ,Â] = 0, where Ĥ is the Hamiltonian operator. + then  is a constant of motion + + Thm: Two operators such that [Â,B̂] always have common Eigenstates \ No newline at end of file diff --git a/lecture_notes/1-27/Derivative of A Operator.jpg b/lecture_notes/1-27/Derivative of A Operator.jpg new file mode 100644 index 0000000..34e25c3 Binary files /dev/null and b/lecture_notes/1-27/Derivative of A Operator.jpg differ diff --git a/lecture_notes/1-27/Expectation Value Time Dependence b/lecture_notes/1-27/Expectation Value Time Dependence index 1811127..874a1a1 100644 --- a/lecture_notes/1-27/Expectation Value Time Dependence +++ b/lecture_notes/1-27/Expectation Value Time Dependence @@ -1,13 +1,16 @@ -Computed Derivative (pic) d/dt (pic) -Discovered/Introduced Commutator -Cpmpatible Observables occur when commutator is equal to 0. +Computed Derivative d/dt : -[Ĥ,Â] = Ĥ - ÂĤ + d = d 〈Ψ(t)|Â|Ψ(t)〉 + dt dt + + ⎛d 〈Ψ(t)⎞ Â|Ψ(t)〉 + 〈Ψ(t)| ⎛d Ψ(t)〉⎞ + 〈Ψ(t)|d Â|Ψ(t)〉 + ⎝dt ⎠ ⎝dt ⎠ dt + ↓ + ι 〈Ψ(t)|Ĥ Â|Ψ(t)〉 + 〈Ψ(t)| Ĥ -ι|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉 + ħ ħ dt + ↓ + ι 〈Ψ(t)|Ĥ  -  Ĥ|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉 + ħ dt -Constant of Motion ------------------- - if d/dt  = 0 and [Ĥ,Â] = 0, - then  is a constant of motion - Thm: Two operators such that [Â,B̂] always have common Eigenstates diff --git a/lecture_notes/1-27/IMG_20160127_131202.jpg b/lecture_notes/1-27/IMG_20160127_131202.jpg new file mode 100644 index 0000000..a399c81 Binary files /dev/null and b/lecture_notes/1-27/IMG_20160127_131202.jpg differ diff --git a/lecture_notes/1-27/IMG_20160127_131402.jpg b/lecture_notes/1-27/IMG_20160127_131402.jpg new file mode 100644 index 0000000..8d8b75b Binary files /dev/null and b/lecture_notes/1-27/IMG_20160127_131402.jpg differ diff --git a/lecture_notes/1-27/IMG_20160127_132453.jpg b/lecture_notes/1-27/IMG_20160127_132453.jpg new file mode 100644 index 0000000..e412558 Binary files /dev/null and b/lecture_notes/1-27/IMG_20160127_132453.jpg differ diff --git a/lecture_notes/1-27/IMG_20160127_132918.jpg b/lecture_notes/1-27/IMG_20160127_132918.jpg new file mode 100644 index 0000000..a0733c3 Binary files /dev/null and b/lecture_notes/1-27/IMG_20160127_132918.jpg differ diff --git a/lecture_notes/1-27/IMG_20160127_133504.jpg b/lecture_notes/1-27/IMG_20160127_133504.jpg new file mode 100644 index 0000000..d0a3fd3 Binary files /dev/null and b/lecture_notes/1-27/IMG_20160127_133504.jpg differ diff --git a/lecture_notes/1-27/IMG_20160127_133506.jpg b/lecture_notes/1-27/IMG_20160127_133506.jpg new file mode 100644 index 0000000..f8db075 Binary files /dev/null and b/lecture_notes/1-27/IMG_20160127_133506.jpg differ diff --git a/lecture_notes/1-27/IMG_20160127_134331.jpg b/lecture_notes/1-27/IMG_20160127_134331.jpg new file mode 100644 index 0000000..0aac1bc Binary files /dev/null and b/lecture_notes/1-27/IMG_20160127_134331.jpg differ diff --git a/solutions/exam1/prob2 b/solutions/exam1/prob2 index fe518f4..8bc6a09 100644 --- a/solutions/exam1/prob2 +++ b/solutions/exam1/prob2 @@ -1,7 +1,24 @@ Show that ͟d͟〈͟p͟〉͟ = -〳͟d͟V͟(͟x͟)͟〵 when a particle is subjected to a potential 〈V(x)〉. - dt 〵 dx 〳 + dt 〵dx 〳 +The time derivative of the expectation value of the momentum is a known quantity, from +Time Dependence of Expectation Value of General Momentum Operator: + +d〈p〉 = 1. +dt ιħ + +The problem is therefore reduced to finding whether -/dV(x)\ reduces to 1. + \dx / ιħ + +-/dV(x)\ = -〈Ψ| d V(x) |Ψ〉. + \dx / dx + + Viewing the expression in this form reveals a relationship between the space derivative and the operators V(x) and |Ψ〉. The chain rule allows this derivative to be computed. + +-〈Ψ| d V(x) |Ψ〉 = -〈Ψ| ⎛d V(x)|Ψ> + d |Ψ> V(x)⎞. + dx ⎝dx dx ⎠ + diff --git a/solutions/exam1/prob2.ps b/solutions/exam1/prob2.ps index 47ff771..55d673a 100644 --- a/solutions/exam1/prob2.ps +++ b/solutions/exam1/prob2.ps @@ -363,6 +363,91 @@ end_ol end_ol } def /JAA { 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 3668 3960 x +2389 4681 m +1713 4908 1360 5394 x +1008 5881 1008 6585 x +1008 7669 1723 8298 x +2439 8928 3668 8928 x +4905 8928 5620 8298 x +6336 7669 6336 6585 x +6336 5887 5989 5400 x +5643 4914 4980 4681 x +5718 4457 6134 3880 x +6552 3304 6552 2497 x +6552 1233 5801 544 x +5051 -144 3668 -144 x +2292 -144 1542 544 x +792 1233 792 2497 x +792 3309 1215 3885 x +1638 4462 2389 4681 x +2592 6453 m +2592 5934 2881 5630 x +3170 5328 3668 5328 x +4173 5328 4462 5630 x +4752 5934 4752 6453 x +4752 6964 4462 7261 x +4173 7560 3668 7560 x +3175 7560 2883 7259 x +2592 6959 2592 6453 x +7200 fwd_x +end_ol + } def +/KAA { start_ol +2664 6264 m +4608 6264 l +4608 4104 l +2664 4104 l +2664 6264 l +2664 2160 m +4608 2160 l +4608 0 l +2664 0 l +2664 2160 l +7200 fwd_x +end_ol + } def +/LAA { start_ol +792 8784 m +6336 8784 l +6336 7554 l +3355 0 l +1584 0 l +4446 7272 l +792 7272 l +792 8784 l +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 2880 4388 m 2880 4695 3088 4903 x 3296 5112 3596 5112 x @@ -393,7 +478,7 @@ end_ol 7200 fwd_x end_ol } def -/KAA { start_ol +/OAA { start_ol 3810 4464 m 3251 4464 2957 4047 x 2664 3631 2664 2840 x @@ -424,41 +509,7 @@ end_ol 7200 fwd_x end_ol } def -/LAA { start_ol -2664 6264 m -4608 6264 l -4608 4104 l -2664 4104 l -2664 6264 l -2664 2160 m -4608 2160 l -4608 0 l -2664 0 l -2664 2160 l -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 +/PAA { start_ol 3668 7344 m 2995 7344 2685 6645 x 2376 5947 2376 4388 x @@ -480,7 +531,7 @@ end_ol 7200 fwd_x end_ol } def -/OAA { start_ol +/QAA { start_ol 4032 8424 m 4032 6552 l 6264 6552 l @@ -503,7 +554,7 @@ end_ol 7200 fwd_x end_ol } def -/PAA { start_ol +/RAA { start_ol 6264 4213 m 6264 0 l 4536 0 l @@ -524,7 +575,7 @@ end_ol 7200 fwd_x end_ol } def -/QAA { start_ol +/SAA { start_ol 3632 5328 m 3012 5328 2658 4785 x 2304 4243 2304 3276 x @@ -546,7 +597,7 @@ end_ol 7200 fwd_x end_ol } def -/RAA { start_ol +/TAA { start_ol 648 3271 m 648 8784 l 2376 8784 l @@ -565,7 +616,7 @@ end_ol 7200 fwd_x end_ol } def -/SAA { start_ol +/UAA { start_ol 2304 2729 m 2304 7848 l 576 7848 l @@ -582,7 +633,7 @@ end_ol 7200 fwd_x end_ol } def -/TAA { start_ol +/VAA { start_ol 6768 4680 m 6487 4968 6111 5112 x 5734 5256 5284 5256 x @@ -603,7 +654,7 @@ end_ol 7200 fwd_x end_ol } def -/UAA { start_ol +/WAA { start_ol 1296 6552 m 4680 6552 l 4680 1296 l @@ -623,7 +674,7 @@ end_ol 7200 fwd_x end_ol } def -/VAA { start_ol +/XAA { start_ol 6264 360 m 5828 108 5328 -18 x 4827 -144 4251 -144 x @@ -646,7 +697,7 @@ end_ol 7200 fwd_x end_ol } def -/WAA { start_ol +/YAA { start_ol 6480 0 m 1008 0 l 1008 8784 l @@ -663,7 +714,7 @@ end_ol 7200 fwd_x end_ol } def -/XAA { start_ol +/ZAA { start_ol 6696 6552 m 4487 3416 l 6912 0 l @@ -680,7 +731,7 @@ end_ol 7200 fwd_x end_ol } def -/YAA { start_ol +/aAA { start_ol 4127 3096 m 3150 3096 2763 2855 x 2376 2614 2376 2030 x @@ -714,7 +765,7 @@ end_ol 7200 fwd_x end_ol } def -/ZAA { start_ol +/bAA { start_ol 4048 5880 m 4242 6296 4540 6495 x 4838 6696 5270 6696 x @@ -746,7 +797,7 @@ end_ol 7200 fwd_x end_ol } def -/aAA { start_ol +/cAA { start_ol 2736 2160 m 4680 2160 l 4680 578 l @@ -757,7 +808,7 @@ end_ol 7200 fwd_x end_ol } def -/bAA { start_ol +/dAA { start_ol 4136 8640 m 3579 6408 l 4771 6408 l @@ -795,7 +846,7 @@ end_ol 7200 fwd_x end_ol } def -/cAA { start_ol +/eAA { start_ol 2664 7344 m 2664 4752 l 3369 4752 l @@ -817,7 +868,7 @@ end_ol 7200 fwd_x end_ol } def -/dAA { start_ol +/fAA { start_ol 4680 3351 m 4680 4214 4351 4735 x 4023 5256 3486 5256 x @@ -852,7 +903,7 @@ end_ol 7200 fwd_x end_ol } def -/eAA { start_ol +/gAA { start_ol 3744 1220 m 3744 1561 3533 1771 x 3323 1982 3011 2050 x @@ -895,7 +946,7 @@ end_ol 4824 fwd_x end_ol } def -/fAA { start_ol +/hAA { start_ol 3456 2284 m 3456 2641 3190 2866 x 2926 3092 2497 3092 x @@ -932,7 +983,7 @@ end_ol 4824 fwd_x end_ol } def -/gAA { start_ol +/iAA { start_ol 2396 3420 m 3162 3420 3688 2896 x 4215 2374 4215 1617 x @@ -954,7 +1005,7 @@ end_ol 4824 fwd_x end_ol } def -/hAA { start_ol +/jAA { start_ol 3524 0 m 3126 0 l 2399 2055 l @@ -984,7 +1035,7 @@ end_ol 4824 fwd_x end_ol } def -/jAA { start_ol +/lAA { start_ol 1479 3312 m 3210 3312 l 3423 3312 3423 3152 x @@ -1014,7 +1065,7 @@ end_ol 4824 fwd_x end_ol } def -/kAA { start_ol +/mAA { start_ol 1000 3025 m 1000 3189 1551 3304 x 2102 3420 2410 3420 x @@ -1051,7 +1102,7 @@ end_ol 4824 fwd_x end_ol } def -/lAA { start_ol +/nAA { start_ol 2418 -536 m 2418 -936 l -2376 -936 l @@ -1060,7 +1111,7 @@ end_ol 0 fwd_x end_ol } def -/mAA { start_ol +/oAA { start_ol 3999 4863 m 3999 327 l 4383 327 l @@ -1091,7 +1142,7 @@ end_ol 4824 fwd_x end_ol } def -/nAA { start_ol +/pAA { start_ol 3240 -835 m 3240 -923 3179 -959 x 3119 -995 3067 -995 x @@ -1109,7 +1160,7 @@ end_ol 4824 fwd_x end_ol } def -/oAA { start_ol +/qAA { start_ol 1119 3312 m 1119 2658 l 1446 3071 1782 3245 x @@ -1144,7 +1195,7 @@ end_ol 4824 fwd_x end_ol } def -/pAA { start_ol +/rAA { start_ol 1584 4651 m 1584 4739 1643 4775 x 1704 4811 1756 4811 x @@ -1162,7 +1213,7 @@ end_ol 4824 fwd_x end_ol } def -/qAA { start_ol +/sAA { start_ol 4171 2664 m 623 2664 l 407 2664 407 2831 x @@ -1180,7 +1231,7 @@ end_ol 4824 fwd_x end_ol } def -/rAA { start_ol +/tAA { start_ol 4003 2088 m 790 2088 l 574 2088 574 2255 x @@ -1191,7 +1242,7 @@ end_ol 4824 fwd_x end_ol } def -/sAA { start_ol +/uAA { start_ol 1767 -936 m 1778 -397 2192 367 x 3735 3267 4651 6162 x @@ -1202,7 +1253,7 @@ end_ol 7992 fwd_x end_ol } def -/tAA { start_ol +/vAA { start_ol 2134 0 m 535 4208 l 288 4208 l @@ -1227,7 +1278,7 @@ end_ol 4824 fwd_x end_ol } def -/uAA { start_ol +/wAA { start_ol 2376 1941 m 2376 2533 2613 3240 x 2851 3947 3116 4387 x @@ -1248,7 +1299,7 @@ end_ol 4824 fwd_x end_ol } def -/vAA { start_ol +/xAA { start_ol 2629 1739 m 4099 327 l 4171 327 l @@ -1289,7 +1340,7 @@ end_ol 4824 fwd_x end_ol } def -/wAA { start_ol +/yAA { start_ol 1224 -831 m 1224 -743 1346 -507 x 1470 -272 1620 11 x @@ -1310,7 +1361,7 @@ end_ol 4824 fwd_x end_ol } def -/xAA { start_ol +/zAA { start_ol 5010 -343 m 3789 2958 2142 5920 x 1802 6529 1767 7056 x @@ -1321,7 +1372,7 @@ end_ol 7992 fwd_x end_ol } def -/yAA { start_ol +/ABA { start_ol 4155 1584 m 831 1584 l 919 948 1357 565 x @@ -1348,7 +1399,7 @@ end_ol 4824 fwd_x end_ol } def -/zAA { start_ol +/BBA { start_ol 2560 3092 m 2415 3092 2282 3064 x 2149 3037 2032 2977 x @@ -1389,7 +1440,7 @@ end_ol 4824 fwd_x end_ol } def -/ABA { start_ol +/CBA { start_ol 4156 2730 m 4117 2730 3909 2893 x 3701 3056 3502 3056 x @@ -1418,7 +1469,7 @@ end_ol 4824 fwd_x end_ol } def -/BBA { start_ol +/DBA { start_ol 2559 3312 m 2559 327 l 3784 327 l @@ -1441,7 +1492,7 @@ end_ol 4824 fwd_x end_ol } def -/CBA { start_ol +/EBA { start_ol 4263 680 m 4263 609 4127 478 x 3991 347 3771 204 x @@ -1471,7 +1522,7 @@ end_ol 4824 fwd_x end_ol } def -/DBA { start_ol +/FBA { start_ol 2559 4863 m 2559 327 l 3784 327 l @@ -1489,7 +1540,7 @@ end_ol 4824 fwd_x end_ol } def -/EBA { start_ol +/GBA { start_ol 3600 928 m 3600 1227 3334 1389 x 3069 1551 2685 1611 x @@ -1531,7 +1582,7 @@ end_ol 4824 fwd_x end_ol } def -/FBA { start_ol +/HBA { start_ol 3528 0 m 3528 525 l 2855 -144 2071 -144 x @@ -1559,7 +1610,7 @@ end_ol 4824 fwd_x end_ol } def -/GBA { start_ol +/IBA { start_ol 1119 4863 m 1119 2601 l 1716 3420 2592 3420 x @@ -1590,7 +1641,7 @@ end_ol 4824 fwd_x end_ol } def -/HBA { start_ol +/JBA { start_ol 3096 2984 m 1233 2984 l 1023 2984 1023 3152 x @@ -1614,7 +1665,7 @@ end_ol 4824 fwd_x end_ol } def -/IBA { start_ol +/KBA { start_ol 2351 939 m 2431 939 l 2662 939 2822 790 x @@ -1627,6 +1678,755 @@ end_ol 1800 643 1959 790 x 2119 939 2351 939 x 4824 fwd_x +end_ol + } def +/LBA { start_ol +2559 327 m +3361 327 l +3567 327 3567 167 x +3567 0 3360 0 x +1431 0 l +1224 0 1224 167 x +1224 327 1429 327 x +2232 327 l +2232 4208 l +903 4208 l +903 3624 l +903 3408 735 3408 x +576 3408 576 3624 x +576 4536 l +4215 4536 l +4215 3624 l +4215 3408 4055 3408 x +3983 3408 3935 3468 x +3888 3529 3888 3624 x +3888 4208 l +2559 4208 l +2559 327 l +4824 fwd_x +end_ol + } def +/MBA { 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 +/NBA { start_ol +2685 0 m +2126 0 l +783 2984 l +455 2984 l +239 2984 239 3152 x +239 3312 455 3312 x +1662 3312 l +1886 3312 1886 3152 x +1886 2984 1662 2984 x +1143 2984 l +2341 327 l +2493 327 l +3668 2984 l +3125 2984 l +2909 2984 2909 3152 x +2909 3312 3125 3312 x +4339 3312 l +4555 3312 4555 3152 x +4555 2984 4339 2984 x +4011 2984 l +2685 0 l +4824 fwd_x +end_ol + } def +/OBA { 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 +/PBA { 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 +/QBA { 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 +/RBA { 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 +/SBA { 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 +/TBA { start_ol +864 327 m +864 4208 l +623 4208 l +432 4208 432 4376 x +432 4536 642 4536 x +2353 4536 l +3108 4536 3625 3943 x +4143 3351 4143 2497 x +4143 2046 l +4143 1184 3625 591 x +3108 0 2353 0 x +642 0 l +432 0 432 167 x +432 327 623 327 x +864 327 l +3816 2562 m +3816 2723 3745 2965 x +3674 3208 3521 3502 x +3368 3796 3049 4002 x +2731 4208 2315 4208 x +1191 4208 l +1191 327 l +2370 327 l +2936 327 3376 826 x +3816 1327 3816 1973 x +3816 2562 l +4824 fwd_x +end_ol + } def +/UBA { 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 +/VBA { start_ol +2511 4316 m +2017 4316 1658 4097 x +1300 3879 1137 3562 x +974 3246 902 2986 x +831 2728 831 2550 x +831 1958 l +831 1131 1324 657 x +1818 183 2677 183 x +3235 183 3816 461 x +3816 1688 l +2730 1688 l +2520 1688 2520 1856 x +2520 2016 2727 2016 x +4216 2016 l +4431 2016 4431 1856 x +4431 1688 4239 1688 x +4143 1688 l +4143 273 l +3426 -144 2653 -144 x +1674 -144 1089 429 x +504 1004 504 1958 x +504 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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 +/YBA { 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 +2351 3312 m +2431 3312 l +2662 3312 2822 3160 x +2982 3008 2982 2784 x +2982 2560 2822 2408 x +2662 2256 2431 2256 x +2351 2256 l +2119 2256 1959 2408 x +1800 2560 1800 2784 x +1800 3008 1959 3160 x +2119 3312 2351 3312 x +4824 fwd_x +end_ol + } def +/ZBA { 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 +/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 +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 +/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 +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 +/eBA { start_ol +1263 5195 m +3816 -367 l +3855 -439 3855 -495 x +3855 -551 3808 -598 x +3762 -646 3699 -646 x +3597 -646 3528 -504 x +975 5059 l +936 5139 936 5186 x +936 5242 986 5289 x +1037 5338 1100 5338 x +1193 5338 1263 5195 x +4824 fwd_x +end_ol + } def +/fBA { start_ol +2823 1440 m +3186 1440 3537 1845 x +3888 2250 3888 2810 x +3888 4536 l +4487 4536 l +4703 4536 4703 4365 x +4703 4216 4487 4216 x +4215 4216 l +4215 2814 l +4215 2440 4098 2130 x +3982 1821 3814 1638 x +3648 1454 3445 1332 x +3243 1210 3088 1165 x +2932 1120 2823 1120 x +2631 1104 l +2631 327 l +2872 327 l +3063 327 3063 167 x +3063 0 2860 0 x +2081 0 l +1872 0 1872 167 x +1872 327 2070 327 x +2304 327 l +2304 1104 l +2124 1112 l +1969 1112 1749 1193 x +1528 1275 1267 1450 x +1006 1625 826 1980 x +648 2334 648 2799 x +648 4208 l +371 4208 l +290 4208 217 4251 x +144 4294 144 4365 x +144 4536 372 4536 x +975 4536 l +975 2793 l +975 2226 1345 1832 x +1716 1440 2116 1440 x +2304 1432 l +2304 4208 l +2142 4208 l +2071 4208 2007 4254 x +1944 4302 1944 4365 x +1944 4536 2155 4536 x +2787 4536 l +2901 4520 2946 4493 x +2991 4466 2991 4365 x +2991 4208 2801 4208 x +2631 4208 l +2631 1432 l +2823 1440 l +4824 fwd_x +end_ol + } def +/gBA { start_ol +2559 4611 m +2559 -775 l +2559 -991 2395 -991 x +2232 -991 2232 -775 x +2232 4611 l +2232 4827 2395 4827 x +2559 4827 2559 4611 x +4824 fwd_x +end_ol + } def +/hBA { start_ol +4119 5482 m +3063 4019 3063 2134 x +3063 -1598 l +2736 -1598 l +2736 1998 l +2736 4091 3820 5641 x +3899 5738 3985 5738 x +4041 5738 4091 5690 x +4143 5641 4143 5577 x +4143 5514 4119 5482 x +4824 fwd_x +end_ol + } def +/iBA { start_ol +623 3964 m +623 4003 667 4071 x +711 4140 783 4140 x +823 4140 887 4099 x +4219 2245 l +878 391 l +815 352 783 352 x +718 352 670 407 x +623 463 623 535 x +623 615 726 671 x +3564 2245 l +726 3820 l +623 3875 623 3964 x +4824 fwd_x +end_ol + } def +/jBA { 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 +/kBA { 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 +/lBA { 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 +/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 end end @@ -1634,14 +2434,32 @@ end end %%EndPrologue %%Page: 1 1 paps_bop -(+ 36000 807000AAABAACAA+ 64800 807000EAAFAAGAA+ 93600 807000HAAIAA+ 115200 807000JAAKAALAAHAAMAALAAHAAMAA+ 180000 807000HAAJAAIAAKAA)paps_exec -(>2950NAAOAAPAAQAA+ 250700 807000RAASAATAAUAAVAAPAALAA+ 308300 807000WAAXAAYAAZAA+ 344300 807000IAAaAA+ 365900 807000bAAHAA)paps_exec -(* 515800cAAYAAdAAFAA+ 551800 807000IAA)paps_exec +(+ 36000 807000AAABAACAA+ 64800 807000EAAFAAGAA+ 93600 807000HAAIAA+ 115200 807000IAAJAAKAAHAALAAKAAMAAMAA+ 180000 807000HAANAAIAAOAA)paps_exec +(>2950PAAQAARAASAA+ 250700 807000TAAUAAVAAWAAXAARAAKAA+ 308300 807000YAAZAAaAAbAA+ 344300 807000IAAcAA+ 365900 807000dAAHAA)paps_exec +(* 515800eAAaAAfAAFAA+ 551800 807000IAA)paps_exec 36 797.000000 moveto 559 797.000000 lineto 0 setlinewidth stroke -(+ 36000 782288eAAfAAgAAhAA+ 60120 782288jAAfAAkAAjAA+ 84240 782288lAAmAAlAAnAAlAAoAAlAApAAlAA+ 108360 782288qAA+ 118008 782288rAAsAAlAAmAAlAAtAAlAAuAAlAAvAAlAAwAAlAAxAA+ 167760 782288hAAfAAyAAzAA+ 191880 782288kAA+ 201528 782288oAAkAAABAjAABBACBADBAyAA+ 244944 782288BBAEBA+ 259416 782288EBAFBAGBAHBAyAACBAjAAyAAmAA+ 307656 782288jAAgAA+ 322128 782288kAA+ 331776 782288oAAgAAjAAyAAzAAjAABBAkAADBA+ 380016 782288nAAtAAuAAvAAwAApAAIBA)paps_exec -(+ 89064 773576mAAjAA+ 122832 773576xAA+ 140472 773576mAAvAA+ 154944 773576sAA)paps_exec +(+ 36000 782288gAAhAAiAAjAA+ 60120 782288lAAhAAmAAlAA+ 84240 782288nAAoAAnAApAAnAAqAAnAArAAnAA+ 108360 782288sAA+ 118008 782288tAAuAAnAAoAAnAAvAAnAAwAAnAAxAAnAAyAAnAAzAA+ 167760 782288jAAhAAABABBA+ 191880 782288mAA+ 201528 782288qAAmAACBAlAADBAEBAFBAABA+ 244944 782288DBAGBA+ 259416 782288GBAHBAIBAJBAABAEBAlAAABAoAA+ 307656 782288lAAiAA+ 322128 782288mAA+ 331776 782288qAAiAAlAAABABBAlAADBAmAAFBA+ 380016 782288pAAvAAwAAxAAyAArAAKBA)paps_exec +(+ 89064 773576oAAlAA+ 122832 773576zAAoAAxAA+ 154944 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620360mBA)paps_exec +()paps_exec ()paps_exec ()paps_exec ()paps_exec diff --git a/solutions/exam1/prob3 b/solutions/exam1/prob3 index 0f8f24f..77762c6 100644 --- a/solutions/exam1/prob3 +++ b/solutions/exam1/prob3 @@ -4,3 +4,5 @@ a) +🔋 +