diff --git a/lecture_notes/1-13/Basis b/lecture_notes/1-13/Basis index 55a0a8f..22145df 100644 --- a/lecture_notes/1-13/Basis +++ b/lecture_notes/1-13/Basis @@ -11,4 +11,4 @@ Quantum State Basis |+x> = C_+ |+z> + C_- |+z> C_+ = e^(i δ_+) / √2 , C_- = e^(i δ_-) / √2 - <+x| S_z |+x> = = expectation value \ No newline at end of file + <+x| S_z |+x> = = expectation value diff --git a/solutions/chap1/prob14/draft b/solutions/chap1/prob14/draft new file mode 100644 index 0000000..e29e530 --- /dev/null +++ b/solutions/chap1/prob14/draft @@ -0,0 +1,11 @@ +c₊²=.36 + + + ___ ___ +|Ψ〉 = √.90 |+z〉 + √.10 |-z〉 + ___ ___ +|Ψ〉 = √.20 |+y〉 + √.80 |-y〉 = √.20 ( ) + ___ ___ ___ ___ +〈Ψ|Ψ〉 = ( 〈+y| √.20 + 〈-y| √.80 ) ( √.90 |+z〉 + √.10 |-z〉 ) + +〈+x|Ψ〉 = d diff --git a/solutions/chap4/draft b/solutions/chap4/draft index 0ba2890..9e4d296 100644 --- a/solutions/chap4/draft +++ b/solutions/chap4/draft @@ -28,7 +28,7 @@ The Hamiltonian is measureable, and therefore is an operator. The magnetic field Therefore, the Hamiltonian Ĥ ≐ -k B𝓏 ħ/2 ( 1 0 ) - ( 0 -1 ) . + ( 0 -1 ) . The commutator can now be computed. This computation is included on an attached sheet. The computation indicates that Ŝ𝓏Ĥ = ĤŜ𝓏, and therefore the commutator is zero. The Hamiltonian and the spin operator in the z direction commute. A similar computation for the x and y directions should indicate a lack of commutability. diff --git a/solutions/chap4/draft.ps b/solutions/chap4/draft.ps index 49337d4..d491e31 100644 --- a/solutions/chap4/draft.ps +++ b/solutions/chap4/draft.ps @@ -178,437 +178,561 @@ end /fontdict 1 dict def papsdict begin fontdict begin /AAA { start_ol -3632 7736 m -2376 3240 l -4890 3240 l -3632 7736 l -2912 8784 m -4359 8784 l -7056 0 l -5823 0 l -5173 2304 l -2086 2304 l -1449 0 l -216 0 l -2912 8784 l +5145 2243 m +1947 2243 l +1298 492 l +2248 492 l +2572 492 2572 252 x +2572 0 2248 0 x +433 0 l +108 0 108 252 x +108 492 433 492 x +805 492 l +2957 6275 l +1515 6275 l +1190 6275 1190 6527 x +1190 6768 1515 6768 x +3967 6768 l +6337 492 l +6781 492 l +7106 492 7106 252 x +7106 0 6781 0 x +4906 0 l +4581 0 4581 252 x +4581 492 4906 492 x +5819 492 l +5145 2243 l +4965 2736 m +3619 6275 l +3439 6275 l +2139 2736 l +4965 2736 l 7200 fwd_x end_ol } def /CAA { start_ol -5688 6264 m -5688 5184 l -5229 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⎜ 0 0 0 ⎟ + ⎝ 0 0 -1 ⎠ + + +To produce the operator S𝓏, one can apply the rotation matrix for a rotation about the y axis., where the angle of rotation is π/2. + + ⎛ cosθ 0 sinθ ⎞ ⎛ 0 0 1 ⎞ + R = ⎜ 0 1 0 ⎟ = ⎜ 0 1 0 ⎟ + ⎝-sinθ 0 cosθ ⎠ ⎝-1 0 0 ⎠ + +S𝓍 = S𝓏 R ≐ + ⎛ 1 0 0 ⎞ ⎛ 0 0 1 ⎞ ⎛ 0 0 1 ⎞ + ħ ⎜ 0 0 0 ⎟ ⎜ 0 1 0 ⎟ = ⎜ 0 1 0 ⎟ + ⎝ 0 0 -1 ⎠ ⎝-1 0 0 ⎠ ⎝-1 0 0 ⎠ + + + + + +The same is true for the S𝓍 operator in its basis. To express this operator in the z basis, however, it must be diagonalized. + + + + + diff --git a/solutions/exam1/prob1.ps b/solutions/exam1/prob1.ps new file mode 100644 index 0000000..3d9f6e8 --- /dev/null +++ b/solutions/exam1/prob1.ps @@ -0,0 +1,2273 @@ +%!PS-Adobe-3.0 +%%Title: Exam 1, Prob 1 - Otho Ulrich +%%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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x +792 3634 1550 4426 x +2309 5220 3372 5220 x +4684 5220 5544 3963 x +5544 6768 l +4919 6768 l +4608 6768 4608 7020 x +4608 7260 4924 7260 x +6036 7260 l +3408 4727 m +2525 4727 1904 4083 x +1284 3441 1284 2518 x +1284 1590 1904 951 x +2525 312 3420 312 x +4303 312 4923 951 x +5544 1590 5544 2495 x +5544 3441 4929 4083 x +4315 4727 3408 4727 x +7200 fwd_x +end_ol + } def +/LBA { start_ol +3390 10 m +1130 4547 l +938 4547 l +613 4547 613 4799 x +613 5040 938 5040 x +2320 5040 l +2644 5040 2644 4799 x +2644 4547 2320 4547 x +1695 4547 l +3667 553 l +5603 4547 l +4953 4547 l +4629 4547 4629 4799 x +4629 5040 4953 5040 x +6276 5040 l +6601 5040 6601 4799 x +6601 4547 6276 4547 x +6108 4547 l +3029 -1739 l +3811 -1739 l +4136 -1739 4136 -1980 x +4136 -2232 3811 -2232 x +1010 -2232 l +685 -2232 685 -1980 x +685 -1739 1010 -1739 x +2536 -1739 l +3390 10 l +7200 fwd_x +end_ol + } def +/MBA { start_ol +5676 5040 m +5676 4606 l +2076 492 l +5400 492 l +5400 1123 l +5400 1428 5652 1428 x +5892 1428 5892 1117 x +5892 0 l +1440 0 l +1440 433 l +5016 4547 l +2004 4547 l +2004 3919 l +2004 3611 1764 3611 x +1512 3611 1512 3925 x +1512 5040 l +5676 5040 l +7200 fwd_x +end_ol + } def +/NBA { start_ol +1716 7260 m +1716 3974 l +2605 5220 3907 5220 x +5011 5220 5775 4444 x +6540 3669 6540 2543 x +6540 1410 5770 615 x +4999 -180 3907 -180 x +2568 -180 1716 1063 x +1716 0 l +547 0 l +216 0 216 263 x +216 492 552 492 x +1224 492 l +1224 6768 l +552 6768 l +216 6768 216 7020 x +216 7260 547 7260 x +1716 7260 l +3882 4727 m +2982 4727 2349 4083 x +1716 3441 1716 2518 x +1716 1596 2349 954 x +2982 312 3882 312 x +4770 312 5409 951 x +6048 1590 6048 2482 x +6048 3429 5421 4078 x +4794 4727 3882 4727 x +7200 fwd_x +end_ol + } def +/OBA { start_ol +1356 5040 m +1356 4414 l +1654 4854 1940 5036 x +2227 5220 2608 5220 x +3380 5220 3777 4353 x +4433 5220 5125 5220 x +5646 5220 6021 4831 x +6396 4443 6396 3910 x +6396 492 l +6798 492 l +7117 492 7117 252 x +7117 0 6793 0 x +5904 0 l +5904 3866 l 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6324 4799 x +6324 4547 5999 4547 x +5915 4547 l +3955 2647 l +7200 fwd_x +end_ol + } def +/RBA { start_ol +3625 6170 m +3877 6170 4052 5989 x +4226 5809 4226 5557 x +4226 5316 4045 5142 x +3865 4968 3624 4968 x +3385 4968 3204 5148 x +3024 5329 3024 5568 x +3024 5809 3204 5989 x +3385 6170 3625 6170 x +6276 3240 m +938 3240 l +613 3240 613 3492 x +613 3732 938 3732 x +6276 3732 l +6601 3732 6601 3492 x +6601 3240 6276 3240 x +6276 1512 m +938 1512 l +613 1512 613 1764 x +613 2004 938 2004 x +6276 2004 l +6601 2004 6601 1764 x +6601 1512 6276 1512 x +7200 fwd_x +end_ol + } def +/SBA { start_ol +6216 8248 m +4596 6048 4596 3210 x +4596 -2405 l +4104 -2405 l +4104 3006 l +4104 6156 5761 8489 x +5880 8633 6013 8633 x +6096 8633 6174 8561 x +6252 8489 6252 8392 x +6252 8296 6216 8248 x +7200 fwd_x +end_ol + } def +/TBA { start_ol +3876 7359 m +3876 492 l +5504 492 l +5820 492 5820 252 x +5820 0 5504 0 x +1768 0 l +1440 0 1440 252 x +1440 492 1766 492 x +3384 492 l +3384 6710 l +2054 5351 l 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l +3096 4194 l +3096 1044 1438 -1289 x +1319 -1433 1186 -1433 x +1103 -1433 1024 -1361 x +947 -1289 947 -1192 x +947 -1096 983 -1048 x +7200 fwd_x +end_ol + } def +/ZBA { start_ol +1328 5040 m +5860 5040 l +6180 5040 6180 4799 x +6180 4547 5862 4547 x +5460 4547 l +5460 492 l +5862 492 l +6180 492 6180 252 x +6180 0 5859 0 x +4425 0 l +4104 0 4104 252 x +4104 492 4428 492 x +4968 492 l +4968 4547 l +2220 4547 l +2220 492 l +2760 492 l +3084 492 3084 252 x +3084 0 2763 0 x +1342 0 l +1008 0 1008 252 x +1008 492 1338 492 x +1728 492 l +1728 4547 l +1326 4547 l +1008 4547 1008 4799 x +1008 5040 1328 5040 x +7200 fwd_x +end_ol + } def +/aBA { start_ol +5761 7611 m +1860 -757 l +1755 -974 1603 -974 x +1508 -974 1437 -901 x +1368 -829 1368 -745 x +1368 -685 1427 -553 x +5328 7815 l +5421 8032 5585 8032 x +5680 8032 5749 7959 x +5820 7887 5820 7804 x +5820 7731 5761 7611 x +7200 fwd_x +end_ol + } def +/bBA { start_ol +1224 5658 m +1224 6201 1885 6820 x +2547 7440 3480 7440 x +4361 7440 5031 6814 x +5700 6188 5700 5369 x +5700 4827 5379 4351 x +5060 3876 3960 2853 x +1458 528 l +1458 492 l +5256 492 l +5256 925 l +5256 1251 5508 1251 x +5748 1251 5748 925 x +5748 0 l +1080 0 l +1080 721 l +3871 3395 l +4699 4225 4953 4599 x +5207 4972 5207 5382 x +5207 6008 4687 6477 x +4169 6948 3480 6948 x +2864 6948 2362 6586 x +1861 6225 1716 5683 x +1642 5442 1458 5442 x +1371 5442 1297 5508 x +1224 5574 1224 5658 x +7200 fwd_x +end_ol + } def +/cBA { start_ol +5820 4226 m +5820 3022 l +5820 1602 5199 711 x +4579 -180 3594 -180 x +2608 -180 1987 711 x +1368 1602 1368 3022 x +1368 4226 l +1368 5658 1987 6549 x +2608 7440 3594 7440 x +4579 7440 5199 6549 x +5820 5658 5820 4226 x +3594 312 m +4402 312 4864 1157 x +5328 2002 5328 3161 x +5328 3395 l +1860 3395 l +1860 3161 l +1860 1953 2341 1132 x +2822 312 3594 312 x +3594 6948 m +2786 6948 2323 6132 x +1860 5316 1860 4197 x +1860 3888 l +5328 3888 l +5328 4197 l +5328 5364 4846 6156 x +4366 6948 3594 6948 x +7200 fwd_x +end_ol + } def 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216000 359576dAAoAAHBAeAAEBAeAAiAAGBA+ 280800 359576kAAjAA+ 302400 359576OBAnAAlAAjAA+ 338400 359576NBAeAA+ 360000 359576KBAkAAhAADBAoAApAAhAAFBAkAAMBAeAAKBACBA)paps_exec +()paps_exec +()paps_exec +()paps_exec +()paps_exec +()paps_exec +paps_eop +showpage +%%Pages: 1 +%%Trailer +%%EOF diff --git a/solutions/exam1/prob2 b/solutions/exam1/prob2 new file mode 100644 index 0000000..414b622 --- /dev/null +++ b/solutions/exam1/prob2 @@ -0,0 +1,47 @@ +͟d͟ ͟\<͟p͟\>͟ = - 〈͟d͟ ͟V͟(͟X͟)͟〉 + dt dx + + If a particle is subject to potential 〈V(x)〉 + +The potential is known, and in the absense of any non-conservative influences, the Hamiltonian is equal to the potential. + +H(x) = V(x) + +H|E〉 = E|E〉 + +Ĥ = ͟p̂͟²͟ + V(x̂) + 2m + +x̂ ≐ x + +p̂ ≐ -ι ħ ͟d͟ + dx + +p̂² ≐ -ħ² ͟d͟² + dx² + +Ĥ = ͟p̂͟²͟ + V(x̂) + 2m + + + ∞ +〈p̂〉 = ∫ dx p(x) = + -∞ + +d/dt ∫ dx p(x) = + + + + +Probably start here: + +〈V(x)〉 = ∫ dx V(x) + +〈d/dx V(x)〉 = ∫ d/dx V(x) dx = V(x) + +d/dt 〈p〉 = d/dt ∫ dx p(x) + +the definition of momentum in function space is + + d/dt p(x) = -d/dx V(x) + diff --git a/solutions/exam1/prob3 b/solutions/exam1/prob3 new file mode 100644 index 0000000..0f8f24f --- /dev/null +++ b/solutions/exam1/prob3 @@ -0,0 +1,6 @@ +|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. + +a) +