From 16180bd6420caaf5369ff8e3c3c2639d726ce818 Mon Sep 17 00:00:00 2001 From: othocaes Date: Mon, 14 Mar 2016 12:41:34 -0400 Subject: [PATCH] finished prob 11 chap 9 --- solutions/chap9/prob11 | 107 +- solutions/chap9/prob11.ps | 4323 ++++++++++++++++++++----------------- 2 files changed, 2398 insertions(+), 2032 deletions(-) diff --git a/solutions/chap9/prob11 b/solutions/chap9/prob11 index 1079d8e..5ea36fa 100644 --- a/solutions/chap9/prob11 +++ b/solutions/chap9/prob11 @@ -39,19 +39,17 @@ The expecation values ❬x̂❭ and ❬p̂❭ are of interest. ⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠ ❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞ x̂ ⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞. - ⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠ ⎝ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠ + ⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠ ⎝ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠ + + + The matrix representations of the position operator x̂ and momentum operator p̂ have been developed from the definition of the increment/decrement operators. The matrix elements may be ascertained by inspection. + + x̂ ≐ √⎛_͟ħ͟ ⎞⎛ 0 √1 ⎞ p̂ ≐ √⎛͟ħ͟m͟ω͟⎞⎛ 0 -ι√1 ⎞ + ⎝2mω⎠⎝ √1 0 ⎠, ⎝ 2 ⎠⎝ ι√1 0 ⎠. - x̂ ≐ p̂ ≐ - ⎛ 0 √1 ⎞ ⎛ 0 -ι√1 ⎞ - ⎝ √1 0 ⎠, ⎝ ι√1 0 ⎠. - - ❬0❙x̂❙0❭ = x₀₀ = ❬1❙x̂❙1❭ = x₁₁ = 0. - - ❬0❙x̂❙1❭ = x₀₁ = ❬1❙x̂❙0❭ = x₁₀ = √1 = 1. - - +ω is a characteristic parameter of the system. It is related to the steepness of the parabolic potential curve. m is the particle's mass. ❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙x̂❙0❭ + ⎞ ⎜ ⎝ ħ ⎠ ⎝ ħ ⎠ ⎟ @@ -64,20 +62,19 @@ The matrix representations of the position operator x̂ and momentum operator p ⎜ ⎟ ⎜ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙x̂❙1❭⎞⎟ ⎝ ⎝ ⎝ ħ 2⎠⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠⎠. + + ❬0❙x̂❙0❭ = x₀₀ = ❬1❙x̂❙1❭ = x₁₁ = 0. + + ❬0❙x̂❙1❭ = x₀₁ = ❬1❙x̂❙0❭ = x₁₀ = √⎛_͟ħ͟ ⎞. + ⎝2mω⎠ Substituting the matrix elements: - ❬x̂❭ = ⅖⎛exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + ⎞ = ⅖⎛exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + ⎞ - ⎜ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎟ ⎜ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎟ - ⎜ ⎟ ⎜ ⎟ - ⎜ exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞⎟ ⎜ exp⎛ι͟⎛⎛E͟₁͟−͟E͟₀͟⎞ - _͟π͟⎞⎞⎟ - ⎝ ⎝ ħ ⎝ ħ 2⎠⎠⎠ ⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠⎠; - - ❬x̂❭ = ⅖⎛ exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ ⎞ - ⎝ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎝ ħ ⎝ ħ 2⎠⎠ ⎠; + ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ ⎞ + ⎝mω⎠⎝ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎝ ħ ⎝ ħ 2⎠⎠ ⎠; - ❬x̂❭ = ⅖⎛ exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛E͟₁͟−͟E͟₀͟⎞t - _͟π͟⎞⎞ ⎞ - ⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎠. + ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛E͟₁͟−͟E͟₀͟⎞t - _͟π͟⎞⎞ ⎞ + ⎝mω⎠⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎠. For the Harmonic Oscillator, Eₙ = ħω(n + ½). @@ -85,15 +82,75 @@ For the Harmonic Oscillator, Eₙ = ħω(n + ½). E₀−E₁ = ħω(0 - 1) = -ħω; E₁−E₀ = ħω. - ❬x̂❭ = ⅖⎛ exp⎛ι⎛⎛−͟ħ͟ω͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞ ⎞ - ⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎠. + ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι⎛⎛−͟ħ͟ω͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞ ⎞ + ⎝mω⎠⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎠. + + ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟)⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - π⎞⎞ ⎞ + ⎝mω⎠ 2 ⎝ ⎝ ⎝ ħ ⎠ ⎠ ⎝ ⎝⎝ ħ⎠ ⎠⎠ ⎠. + + ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟)⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ - exp⎛ι⎛ħ͟ω͟⎞t⎞ ⎞ + ⎝mω⎠ 2 ⎝ ⎝ ⎝ ħ ⎠ ⎠ ⎝ ⎝ ħ⎠ ⎠ ⎠. + +This is a sine function. + + ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟) 2ι sin⎛−͟ħ͟ω͟t͟⎞ = ⅕ √⎛͟8͟ħ͟⎞ ι² -sin(ωt). + ⎝mω⎠ 2 ⎝ ħ ⎠ ⎝mω⎠ (𝐜,x̂) - ❬x̂❭ = ⅖⎛ ι(-ωt + _͟π͟)⎞ + exp⎛ι⎛ωt - _͟π͟⎞⎞ ⎞ - ⎝ e 2 ⎠ ⎝ ⎝ 2⎠⎠ ⎠. + ❬x̂❭ = ⅕ √⎛͟8͟ħ͟⎞ sin(ωt). + ⎝mω⎠ -The expectation value progresses with the time parameter t. ω is a characteristic parameter of the system. It is related to the steepness of the parabolic potential curve. +The expectation value progresses periodically with the time parameter t. + + +A very similar argument can be made for the momentum operator. + + ❬p̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙p̂❙0❭ + ⎞ + ⎜ ⎝ ħ ⎠ ⎝ ħ ⎠ ⎟ + ⎜ ⎟ + ⎜ exp⎛ι͟E͟₀͟t⎞ 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞ ❬0❙p̂❙1❭ + ⎟ + ⎜ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎟ + ⎜ ⎟ + ⎜ exp⎛-ι͟E͟₀͟t⎞ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞ ❬1❙p̂❙0❭ + ⎟ + ⎜ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎟ + ⎜ ⎟ + ⎜ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙p̂❙1❭⎞⎟ + ⎝ ⎝ ⎝ ħ 2⎠⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠⎠. + ❬0❙p̂❙0❭ = p₀₀ = ❬1❙p̂❙1❭ = p₁₁ = 0. + + ❬0❙p̂❙1❭ = p₀₁ = -ι√⎛͟ħ͟m͟ω͟⎞. + ⎝ 2 ⎠ + + ❬1❙p̂❙0❭ = p₁₀ = ι√⎛͟ħ͟m͟ω͟⎞. + ⎝ 2 ⎠ + + ❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞⎛-exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ + ⎝ 2 ⎠⎜ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎝ ħ ⎝ ħ 2⎠⎠. + + ❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞⎛exp⎛ι⎛⎛−͟ħ͟ω͟⎞t - _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞⎞ + ⎝ 2 ⎠⎝ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠⎠. + + ❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞ exp⎛−͟ι͟π͟⎞ ⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ + exp⎛ι⎛ħ͟ω͟⎞t⎞ ⎞ + ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ⎝ ⎝ ħ⎠ ⎠ ⎝ ⎝ ħ⎠ ⎠ ⎠. + +This is a cosine. + +(𝐜,p̂) + + ❬p̂❭ = ⅘ ι √⎛͟ħ͟m͟ω͟⎞ exp⎛−͟ι͟π͟⎞ cos(ωt) = ⅕ √(8ħmω) cos(ωt). + ⎝ 2 ⎠ ⎝ 2 ⎠ + + +Ehrenfest's theorem states + + ❬p̂❭ = m d͟❬͟x̂͟❭͟. + dt + + m d͟❬͟x̂͟❭͟ = ⅕ mω √⎛͟8͟ħ͟⎞ cos(ωt) = ⅕ √(8mωħ) cos(ωt) = ❬p̂❭. + dt ⎝mω⎠ + +So, the theorem holds for this case. diff --git a/solutions/chap9/prob11.ps b/solutions/chap9/prob11.ps index c6f23c6..96bf6b2 100644 --- a/solutions/chap9/prob11.ps +++ b/solutions/chap9/prob11.ps @@ -1,5 +1,5 @@ %!PS-Adobe-3.0 -%%Title: prob11 +%%Title: Chap9 #11: Otho Ulrich %%Creator: paps version 0.6.7 by Dov Grobgeld %%Pages: (atend) %%BoundingBox: 0 0 595 841 @@ -361,6 +361,37 @@ end_ol end_ol } def /JAA { 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 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+1046 4061 991 4110 x +936 4160 936 4223 x +5400 fwd_x end_ol } def /MBA { 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 +4545 490 m +4545 415 4410 273 x +4275 131 3982 -5 x +3690 -144 3357 -144 x +2907 -144 2641 173 x +2376 490 2376 999 x +2376 3375 l +1827 3375 l +1584 3375 1584 3559 x +1584 3744 1827 3744 x +2745 3744 l +2745 1003 l +2745 225 3357 225 x +3591 225 3775 291 x +3960 358 4050 441 x +4140 523 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/WBA { start_ol -1183 3244 m -1183 3206 1291 3042 x -1399 2877 1399 2747 x -1399 2549 1251 2426 x -1102 2304 868 2304 x -625 2304 492 2436 x -360 2570 360 2806 x -360 3234 779 3501 x -1200 3767 1887 3767 x -2619 3767 3048 3477 x -3478 3187 3478 2693 x -3478 618 l -3478 371 3646 371 x -3702 371 3733 417 x -3813 531 l -3934 318 l -3510 -108 3038 -108 x -2742 -108 2595 21 x -2448 151 2383 461 x -1756 -108 1095 -108 x -715 -108 465 136 x -216 380 216 753 x -216 1000 348 1210 x -481 1422 658 1557 x -835 1692 1145 1827 x -1455 1962 1672 2032 x -1891 2104 2245 2208 x -2335 2232 2385 2247 x -2385 2828 l -2385 3193 2221 3360 x -2058 3528 1699 3528 x -1487 3528 1335 3447 x -1183 3367 1183 3244 x -2376 817 m -2376 1969 l -1382 1715 1382 1008 x -1382 769 1496 622 x -1611 475 1797 475 x -2074 475 2376 817 x -4032 fwd_x +3960 0 m +3960 592 l +3211 -144 2340 -144 x +1804 -144 1478 178 x +1152 502 1152 1031 x +1152 3375 l +671 3375 l +432 3375 432 3564 x +432 3744 672 3744 x +1521 3744 l +1521 1032 l 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fwd_x end_ol } def /aBA { start_ol -3528 1487 m -3528 0 l -3096 0 l -3096 1464 l -3096 1812 2938 1985 x -2782 2160 2468 2160 x -2090 2160 1873 1950 x -1656 1741 1656 1382 x -1656 0 l -1224 0 l -1224 2448 l -1656 2448 l -1656 2070 l -1818 2288 2040 2404 x -2264 2520 2556 2520 x -3034 2520 3280 2256 x -3528 1993 3528 1487 x -4824 fwd_x +3096 7200 m +3465 7200 l +3465 -1800 l +3096 -1800 l +3096 7200 l +5400 fwd_x end_ol } def /bBA { start_ol -1656 1158 m -2718 1158 l -1560 -958 l -1455 -1158 1311 -1158 x -1216 -1158 1147 -1090 x -1080 -1022 1080 -927 x -1080 -903 1096 -839 x -1656 1158 l -2135 3312 m -2215 3312 l -2446 3312 2606 3160 x -2766 3008 2766 2784 x -2766 2560 2606 2408 x -2446 2256 2215 2256 x -2135 2256 l -1903 2256 1743 2408 x -1584 2560 1584 2784 x -1584 3008 1743 3160 x -1903 3312 2135 3312 x -4824 fwd_x +2385 -1800 m +2016 -1800 l +2016 7200 l +2385 7200 l +2385 -1800 l +5400 fwd_x end_ol } def /cBA { start_ol -3456 2284 m -3456 2641 3190 2866 x -2926 3092 2497 3092 x 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l +1296 1872 l +3384 1872 l +5400 fwd_x end_ol } def /dBA { start_ol -1686 5375 m -1686 3368 l -2121 3816 2687 3816 x -3354 3816 3776 3290 x -4198 2765 4198 1923 x -4198 1024 3690 457 x -3182 -108 2376 -108 x -1823 -108 1302 334 x -671 -100 l -576 -100 l -576 4722 l -576 4961 497 5044 x -418 5128 144 5184 x -144 5375 l -1686 5375 l -1686 2905 m -1686 772 l -1686 480 1841 306 x -1998 131 2252 131 x -3024 131 3024 1778 x -3024 3384 2261 3384 x -1875 3384 1686 2905 x -4464 fwd_x +4509 2304 m +891 2304 l +648 2304 648 2493 x +648 2673 891 2673 x +4509 2673 l +4752 2673 4752 2493 x +4752 2304 4509 2304 x +5400 fwd_x end_ol } def /eBA { start_ol -3263 148 m -3263 -308 l -3263 -852 3022 -1183 x -2783 -1515 2399 -1515 x -2016 -1515 1775 -1183 x -1535 -852 1535 -308 x -1535 148 l -1535 685 1771 988 x -2008 1292 2399 1292 x -2791 1292 3027 988 x -3263 685 3263 148 x -2595 981 m -2512 1004 2399 1004 x -2287 1004 2207 981 x -2127 957 2027 878 x -1927 799 1875 607 x -1823 415 1823 124 x -1823 -276 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