diff --git a/HW b/HW index 31f7e4a..ece59bb 100644 --- a/HW +++ b/HW @@ -6,5 +6,5 @@ Chap 5: 2, 6, 8, 12, in-class assignment Chap 9: 7, 11, 12, 13, 14 -Chap 7: 7, 8, one from last class, 11 +Chap 7: 5, 7, 8, one from last class, 11 due friday 3-18, the class one is: show L̂𝓏 and p² commute diff --git a/lecture_notes/3-14/3d eigenstates b/lecture_notes/3-14/3d eigenstates index 5e9fb7f..1ca4739 100644 --- a/lecture_notes/3-14/3d eigenstates +++ b/lecture_notes/3-14/3d eigenstates @@ -55,4 +55,4 @@ Understanding a System: With the eigenvalue equation - ❬r❙E,l,mₗ❭ = E❬r❙E,l,mₗ❭ \ No newline at end of file + ❬r❙E,l,mₗ❭ = E❬r❙E,l,mₗ❭ diff --git a/lecture_notes/3-14/rotational invariance b/lecture_notes/3-14/rotational invariance index 3f07e53..a14e8cb 100644 --- a/lecture_notes/3-14/rotational invariance +++ b/lecture_notes/3-14/rotational invariance @@ -42,7 +42,7 @@ Rotational Invariance = p̂𝓍[L̂𝓏,p̂𝓍] + [L̂𝓏,p̂𝓍]p̂𝓍 + p̂𝓎[L̂𝓏,p̂𝓎] + [L̂𝓏,p̂𝓎]p̂𝓎 - + p̂𝓏[L̂𝓏,p̂𝓏] + [L̂𝓏,p̂𝓏]p̂𝓏 (f.n. 1) + + p̂𝓏[L̂𝓏,p̂𝓏] + [L̂𝓏,p̂𝓏]p̂𝓏 note¹ = ιħp̂𝓍p̂𝓎 + ιħp̂𝓎p̂𝓍 - ιħp̂𝓎p̂𝓍 - ιħp̂𝓍p̂𝓎 = 0 @@ -69,7 +69,7 @@ Rotational Invariance -(f.n. 1) +note¹ [L̂𝓏,p̂𝓍²] = [L̂𝓏,p̂𝓍 p̂𝓍] diff --git a/lecture_notes/3-23/overview b/lecture_notes/3-23/overview index 9781edc..cda461c 100644 --- a/lecture_notes/3-23/overview +++ b/lecture_notes/3-23/overview @@ -1,7 +1,7 @@ Two dimensional harmonic oscillator ─────────────────────────────────── - This is an oscillator with potential V(x,y) = μ/2 ω (x² + y²) + This is an oscillator with potential V(x,y) = μ/2 ω² (x² + y²) The hamiltonian here leaves us with a 3-dimensional differential equation @@ -14,13 +14,14 @@ Two dimensional harmonic oscillator Put together solutions of Ψ(x,y) and Ψ(z). - (pic) To find position space representation of Ψ(x,y), recall the Hermitian Polynomials solution + (pic) To find position space representation of Ψ(x,y), recall the Hermitian-- HARMONIC? DUBIOUS Polynomials solution !!! STUDY THIS !!! Developed the harmonic oscillator in polar coordinates - Ĥ𝓍𝓎 = -ħ²\2μ ∇² + μ/2 ω² r² + Ĥ𝓍𝓎 = -ħ²/2μ ∇² + μ/2 ω² r² + = -ħ²/2μ (∂²/∂r² + 1/r ∂/∂r + 1/r² ∂²/∂θ²) + μ/2 ω² r². (pic) Can be solved using separation of variables. @@ -32,4 +33,10 @@ Two dimensional harmonic oscillator Use L̂𝓏² Θ = -m²ħ² Θ ⇒ Θ(θ) = exp(±imθ) - (pic) further developed hamiltonian using this information \ No newline at end of file + (pic) further developed hamiltonian using this information + + Ĥ𝓍𝓎 = -ħ²/2μ (∂²/∂r² + 1/r ∂/∂r + 1/r² ∂²/∂θ²) + μ/2 ω² r². + = -ħ²/2μ (∂²/∂r² + 1/r ∂/∂r) + -ħ²/(2μr²) L̂𝓏²/ħ² + μ ω² r²/2 + + L̂𝓏² ≐ ħ²∂²/∂θ² + L̂𝓏 ≐ -ιħ∂/∂θ \ No newline at end of file diff --git a/solutions/chap7/lectureprob b/solutions/chap7/lectureprob new file mode 100644 index 0000000..a0aeb12 --- /dev/null +++ b/solutions/chap7/lectureprob @@ -0,0 +1,21 @@ +This problem associated with chapter 7 was assigned during lecture. + +Does L̂𝓏 commute with 𝐫̂²? + + [L̂𝓏,𝐫̂²] = L̂𝓏 𝐫̂² - 𝐫̂² L̂𝓏. + + L̂𝓏 𝐫̂² - 𝐫̂² L̂𝓏. + +Using the position representations, in spherical coordinates, + + L̂𝓏 ≐ -ιħ∂/∂θ and 𝐫̂² ≐ 𝐫², + + L̂𝓏 𝐫̂² - 𝐫̂² L̂𝓏 = 𝐫² ιħ∂/∂θ - ιħ∂/∂θ 𝐫². + +𝐫² has no θ dependence, so it can be separated from any quantity differentiated with respect to theta, I.E., + + ∂/∂θ 𝐫² = 𝐫² ∂/∂θ. + + 𝐫² ιħ∂/∂θ - ιħ∂/∂θ 𝐫² = 𝐫² ιħ∂/∂θ - 𝐫² ιħ∂/∂θ = 0 = [L̂𝓏,𝐫̂²] = 0. + + [L̂𝓏,𝐫̂²] = 0, so these quantities commute. \ No newline at end of file diff --git a/solutions/exam1/prob2.ps b/solutions/chap7/lectureprob.ps similarity index 64% rename from solutions/exam1/prob2.ps rename to solutions/chap7/lectureprob.ps index 55d673a..57bb532 100644 --- a/solutions/exam1/prob2.ps +++ b/solutions/chap7/lectureprob.ps @@ -1,5 +1,5 @@ %!PS-Adobe-3.0 -%%Title: Otho Ulrich: Exam 1, #2 +%%Title: Otho Ulrich: Class Prob %%Creator: paps version 0.6.7 by Dov Grobgeld %%Pages: (atend) %%BoundingBox: 0 0 595 841 @@ -178,96 +178,24 @@ end /fontdict 1 dict def papsdict begin fontdict begin /AAA { start_ol -3058 3891 m -1732 4386 1262 4930 x -792 5475 792 6395 x -792 7576 1553 8251 x -2315 8928 3646 8928 x -4249 8928 4852 8781 x -5456 8636 6048 8352 x -6048 6696 l -5496 7083 4927 7285 x -4359 7488 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l +2962 2531 3280 2810 x +3600 3090 3600 3439 x +3600 3827 3265 4071 x +2932 4316 2415 4316 x +1938 4316 1407 4053 x +1407 3737 l +1407 3528 1247 3528 x +1080 3528 1080 3737 x +1080 4262 l +2255 690 m +2471 690 l +2679 690 2803 578 x +2927 466 2927 291 x +2927 115 2799 3 x +2671 -108 2471 -108 x +2255 -108 l +2047 -108 1923 3 x +1800 115 1800 291 x +1800 466 1923 578 x +2047 690 2255 690 x 4824 fwd_x end_ol } def -/XBA { start_ol -2396 4644 m -3232 4644 3795 3944 x -4359 3244 4359 2217 x -4359 1238 3783 546 x -3208 -144 2396 -144 x -1576 -144 1003 550 x -432 1245 432 2250 x -432 3254 1003 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 +/GBA { start_ol +2559 4536 m +2559 -680 l +3291 -680 l +3495 -680 3495 -840 x +3495 -1008 3288 -1008 x +2232 -1008 l +2232 4863 l +3288 4863 l +3495 4863 3495 4703 x +3495 4536 3291 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+The probability for is given by + + │❬1 m′❙Ψ❭│², with m′ = {-1, 0, 1}. + +The eigenstates form an orthogonal set such that + + ❬l′ m′❙l m❭ = δₗₗ′ δₘₘ′. + +Then, + + ❬1 1❙Ψ❭ = ❬1 1❙⎛ 2͟ ❙1 1❭ + ι 3͟ ❙1 0❭ - 4͟ ❙1 -1❭ ⎞ + ⎝√29 √29 √29 ⎠ + + = ❬1 1❙ 2͟ ❙1 1❭ = 2͟ . + √29 √29 + +(𝐚) + │❬1 1❙Ψ❭│² = 4͟ = ⁴/₂₉. + 29 + +Similarly, + + │❬1 0❙Ψ❭│² = 9͟ = ⁹/₂₉ and + 29 + + │❬1 -1❙Ψ❭│² = 1͟6͟ = ¹⁶/₂₉. + 29 + + +The eigenvalue equations for the L̂𝓏 operator are simplified because L̂𝓏 is diagonal in the z basis. The L̂𝓍 operator produces the same measurements, but the matrix representation of the L̂𝓍 operator must be applied. It is + + L̂𝓍 ≐ + ħ͟ ⎛ 0 1 0 ⎞ + √2 ⎜ 1 0 1 ⎟ + ⎝ 0 1 0 ⎠. + +Applying the operator to the states in Ψ, + + L̂𝓍❙1 1❭ ≐ + ħ͟ ⎛ 0 1 0 ⎞⎛1⎞ = ħ͟ ⎛0⎞ = ħ͟ ❙1 0❭. + √2 ⎜ 1 0 1 ⎟⎜0⎟ √2 ⎜1⎟ √2 + ⎝ 0 1 0 ⎠⎝0⎠ ⎝0⎠ + + L̂𝓍❙1 0❭ ≐ + ħ͟ ⎛ 0 1 0 ⎞⎛0⎞ = ħ͟ ⎛1⎞ = ħ͟ (❙1 1❭ + ❙1 -1❭), and + √2 ⎜ 1 0 1 ⎟⎜1⎟ √2 ⎜0⎟ √2 + ⎝ 0 1 0 ⎠⎝0⎠ ⎝1⎠ + + L̂𝓍❙1 -1❭ ≐ + ħ͟ ⎛ 0 1 0 ⎞⎛0⎞ = ħ͟ ⎛0⎞ = ħ͟ ❙1 0❭. + √2 ⎜ 1 0 1 ⎟⎜0⎟ √2 ⎜1⎟ √2 + ⎝ 0 1 0 ⎠⎝1⎠ ⎝0⎠ + + + L̂𝓍❙Ψ❭ = ⎛ 2͟ L̂𝓍❙1 1❭ + ι 3͟ L̂𝓍❙1 0❭ - 4͟ L̂𝓍❙1 -1❭ ⎞ + ⎝ √29 √29 √29 ⎠ + + 2͟ L̂𝓍❙1 1❭ = 2͟ ħ❙1 0❭, + √29 √58 + + ι 3͟ L̂𝓍❙1 0❭ = ι 3͟ ħ (❙1 1❭ + ❙1 -1❭), and + √29 √58 + + 4͟ L̂𝓍❙1 -1❭ = 4͟ ħ❙1 0❭. + √29 √58 + +Then, + + L̂𝓍❙Ψ❭ = ħ⎛ -2͟ ❙1 0❭ + ι 3͟ (❙1 1❭ + ❙1 -1❭)⎞ + ⎝ √58 √58 ⎠ + +Normalizing the function, + + C⎛⎛-2͟ ⎞² + ⎛ι 3͟ ⎞² + ⎛ι 3͟ ⎞²⎞ = 1. + ⎝⎝√58⎠ ⎝ √58⎠ ⎝ √58⎠ ⎠ + + + STOPPED HERE + + C = 58⎛⎛1͟⎞ - ⎛ι 3 ⎞⁻² + ⎛ι 3 ⎞⁻²⎞ + ⎝⎝4⎠ ⎝ ⎠ ⎝ ⎠ ⎠ +So, + + \|❬1 1❙L̂𝓍❙Ψ❭\|^2 = + + + + + + L̂𝓍❙Ψ❭ = ⎛ + ι 3͟ L̂𝓍❙1 0❭ - 4͟ L̂𝓍❙1 -1❭ ⎞ + ⎝ √29 √29 ⎠ + + + diff --git a/solutions/chap7/prob5.ps b/solutions/chap7/prob5.ps new file mode 100644 index 0000000..1f9f504 --- /dev/null +++ b/solutions/chap7/prob5.ps @@ -0,0 +1,3448 @@ +%!PS-Adobe-3.0 +%%Title: Otho Ulrich: Chap 7 #5 +%%Creator: paps version 0.6.7 by Dov Grobgeld +%%Pages: (atend) +%%BoundingBox: 0 0 595 841 +%%BeginProlog +%%Orientation: Portrait +/papsdict 1 dict def +papsdict begin + +/inch {72 mul} bind def +/mm {1 inch 25.4 div mul} bind def + +% override setpagedevice if it is not defined +/setpagedevice where { + pop % get rid of its dictionary + /setpagesize { + 3 dict begin + /pageheight exch def + /pagewidth exch def + /orientation 0 def + % Exchange pagewidth and pageheight so that pagewidth is bigger + pagewidth pageheight gt { + pagewidth + /pagewidth pageheight def + /pageheight exch def + /orientation 3 def + } if + 2 dict + dup /PageSize [pagewidth pageheight] put + dup /Orientation orientation put + setpagedevice + end + } def +} +{ + /setpagesize { pop pop } def +} ifelse +/duplex { + statusdict /setduplexmode known + { statusdict begin setduplexmode end } {pop} ifelse +} def +/tumble { + statusdict /settumble known + { statusdict begin settumble end } {pop} ifelse +} def +% Turn the page around +/turnpage { + 90 rotate + 0 pageheight neg translate +} def +% User settings +/pagewidth 595 def +/pageheight 841 def +pagewidth pageheight setpagesize +/column_width 523 def +/bodyheight 755 def +/lmarg 36 def +/ytop 791 def +/do_separation_line true def +/do_landscape false def +/do_tumble true def +/do_duplex true def +% Procedures to translate position to first and second column +/lw 20 def % whatever +/setnumcolumns { + /numcolumns exch def + /firstcolumn { /xpos lmarg def /ypos ytop def} def + /nextcolumn { + do_separation_line { + xpos column_width add gutter_width 2 div add % x start + ytop lw add moveto % y start + 0 bodyheight lw add neg rlineto 0 setlinewidth stroke + } if + /xpos xpos column_width add gutter_width add def + /ypos ytop def + } def +} def + +1 setnumcolumns +/showline { + /y exch def + /s exch def + xpos y moveto + column_width 0 rlineto stroke + xpos y moveto /Helvetica findfont 20 scalefont setfont s show +} def +/paps_bop { % Beginning of page definitions + papsdict begin + gsave + do_landscape {turnpage} if + % ps2pdf gets wrong orientation without this! 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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 +/vAA { 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 +/xAA { start_ol +3008 5592 m +3008 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start_ol +2418 -536 m +2418 -936 l +-2376 -936 l +-2376 -536 l +2418 -536 l +0 fwd_x +end_ol + } def +/DBA { start_ol +2559 4927 m +2559 327 l +3645 327 l +3855 327 3855 167 x +3855 0 3645 0 x +1154 0 l +936 0 936 167 x +936 327 1153 327 x +2232 327 l +2232 4495 l +1381 3592 l +1320 3528 1222 3528 x +1163 3528 1121 3579 x +1080 3632 1080 3712 x +1080 3784 1164 3872 x +2170 4927 l +2559 4927 l +4824 fwd_x +end_ol + } def +/EBA { start_ol +2559 2088 m +2559 500 l +2559 288 2399 288 x +2232 288 2232 500 x +2232 2088 l +855 2088 l +648 2088 648 2255 x +648 2415 855 2415 x +2232 2415 l +2232 4002 l +2232 4215 2399 4215 x +2559 4215 2559 4002 x +2559 2415 l +3992 2415 l +4215 2415 4215 2255 x +4215 2088 3992 2088 x +2559 2088 l +4824 fwd_x +end_ol + } def +/FBA { start_ol +3999 456 m +3999 385 3880 250 x +3762 117 3503 -13 x +3245 -144 2952 -144 x +2556 -144 2322 137 x +2088 419 2088 871 x +2088 2984 l +1639 2984 l +1440 2984 1440 3147 x +1440 3312 1643 3312 x +2415 3312 l +2415 876 l +2415 183 2954 183 x +3160 183 3322 249 x +3484 317 3562 399 x +3642 481 3718 547 x +3793 615 3840 615 x +3864 615 3900 603 x +3936 591 3967 552 x +3999 513 3999 456 x +4824 fwd_x +end_ol + } def +/GBA { start_ol +970 4285 m +970 4368 1126 4521 x +1282 4674 1625 4822 x +1969 4971 2377 4971 x +2976 4971 3379 4614 x +3783 4258 3783 3737 x +3783 3272 3523 3000 x +3263 2728 2959 2648 x +3431 2454 3714 2112 x +3999 1771 3999 1361 x +3999 751 3507 321 x +3016 -108 2328 -108 x +1857 -108 1317 128 x +778 364 778 564 x +778 621 825 669 x +874 717 930 717 x +986 717 1154 592 x +1321 468 1633 343 x +1945 219 2336 219 x +2880 219 3276 556 x +3672 895 3672 1361 x +3672 1836 3245 2178 x +2820 2520 2233 2520 x +2016 2520 2016 2687 x +2016 2847 2227 2847 x +3456 2847 3456 3729 x +3456 4123 3144 4383 x +2832 4644 2369 4644 x +2034 4644 1797 4566 x +1562 4488 1470 4392 x +1378 4295 1290 4217 x +1202 4140 1130 4140 x +1066 4140 1018 4181 x +970 4224 970 4285 x +4824 fwd_x +end_ol + } def +/HBA { start_ol +3855 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x +2936 4900 2936 5027 x +2936 5234 2767 5388 x +2598 5544 2371 5544 x +2167 5544 2007 5428 x +1847 5313 1792 5131 x +1760 5004 1648 5004 x +1591 5004 1551 5043 x +1512 5083 1512 5139 x +4824 fwd_x +end_ol + } def +/fBA { start_ol +1263 4160 m +1263 3903 1571 3735 x +1879 3567 2358 3567 x +3151 3567 3750 2889 x +4143 2454 4143 1743 x +4143 1011 3634 469 x +3127 -72 2343 -72 x +1613 -72 1094 469 x +576 1011 576 1743 x +576 2488 949 2876 x +1402 3336 1657 3407 x +1359 3454 1147 3656 x +936 3858 936 4143 x +936 4595 1212 4837 x +1489 5079 1949 5079 x +2365 5079 3600 4759 x +3600 4944 l +3600 5151 3816 5151 x +3927 5151 3927 4939 x +3927 4330 l +3927 4104 3816 4104 x +3600 4104 3600 4333 x +3600 4443 l +2412 4752 1973 4752 x +1649 4752 1455 4600 x +1263 4448 1263 4160 x +2343 255 m +2999 255 3407 696 x +3816 1138 3816 1743 x +3816 2077 3695 2352 x +3576 2626 3399 2785 x +3224 2945 3011 3052 x +2800 3160 2631 3199 x +2463 3240 2343 3240 x +1732 3240 1317 2805 x +903 2372 903 1743 x +903 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- } def -/VBA { 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 -/WBA { start_ol -3384 8006 m -3794 8006 l -3794 -2001 l -3384 -2001 l -3384 8006 l -5976 fwd_x -end_ol - } def -/XBA { start_ol -2642 -2001 m -2232 -2001 l -2232 8006 l -2642 8006 l -2642 -2001 l -5976 fwd_x -end_ol - } def -/YBA { start_ol -5133 -860 m -5162 -901 5162 -981 x -5162 -1060 5097 -1120 x -5033 -1181 4963 -1181 x -4914 -1181 4864 -1150 x -4815 -1120 4784 -1081 x -4755 -1050 l -3384 870 3384 3503 x -3384 8006 l -3794 8006 l -3794 3332 l -3794 970 5133 -860 x -5976 fwd_x -end_ol - } def -/ZBA { start_ol -842 -889 m -2181 941 2181 3304 x -2181 7977 l -2592 7977 l -2592 3474 l -2592 851 1220 -1090 x -1121 -1209 1012 -1209 x -942 -1209 877 -1149 x -813 -1090 813 -1009 x -813 -929 842 -889 x -5976 fwd_x 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