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still working on exam 1

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34
solutions/exam1/prob1
View File

@ -4,36 +4,38 @@ The spin eigenstates and eigenvalues are known from experiment for a spin-1 syst
S𝓏 ≐ S𝓏 ≐
⎛ 1 0 0 ⎞ ħ ⎛ 1 0 0 ⎞
ħ ⎜ 0 0 0 ⎟ and ⎜ 0 0 0 ⎟ and
⎝ 0 0 -1 ⎠ ⎝ 0 0 -1 ⎠
S𝓍 ≐ S𝓍 ≐
⎛ 0 1 0 ⎞ ͟ħ͟ ⎛ 0 1 0 ⎞
ħ ⎜ 1 0 1 ⎟. √2 ⎜ 1 0 1 ⎟.
⎝ 0 1 0 ⎠ ⎝ 0 1 0 ⎠
Using the matrix representations, the expressions can be evaluated. For the spin-z operator, the expression S𝓏 (S𝓏 + ħ)(S𝓏 - ħ) ≐ Using the matrix representations, the expressions can be evaluated. For the spin-z operator, the expression S𝓏 (S𝓏 + ħ)(S𝓏 - ħ) ≐
⎛ 1 0 0 ⎞ ⎧ ⎛ 1 0 0 ⎞ ⎛ 1 0 0 ⎞ ⎫ ⎧ ⎛ 1 0 0 ⎞ ⎛ 1 0 0 ⎞ ⎫ ħ ⎛ 1 0 0 ⎞ ⎧ ħ ⎛ 1 0 0 ⎞ ħ ⎛ 1 0 0 ⎞ ⎫ ⎧ ħ ⎛ 1 0 0 ⎞ ħ ⎛ 1 0 0 ⎞ ⎫
ħ ⎜ 0 0 0 ⎟ ⎪ ħ ⎜ 0 0 0 ⎟ + ħ ⎜ 0 1 0 ⎟ ⎪ ⎪ ħ ⎜ 0 0 0 ⎟ - ħ ⎜ 0 1 0 ⎟ ⎪, ⎜ 0 0 0 ⎟ ⎪ ⎜ 0 0 0 ⎟ + ⎜ 0 1 0 ⎟ ⎪ ⎪ ⎜ 0 0 0 ⎟ - ⎜ 0 1 0 ⎟ ⎪,
⎝ 0 0 -1 ⎠ ⎩ ⎝ 0 0 -1 ⎠ ⎝ 0 0 1 ⎠ ⎭ ⎩ ⎝ 0 0 -1 ⎠ ⎝ 0 0 1 ⎠ ⎭ ⎝ 0 0 -1 ⎠ ⎩ ⎝ 0 0 -1 ⎠ ⎝ 0 0 1 ⎠ ⎭ ⎩ ⎝ 0 0 -1 ⎠ ⎝ 0 0 1 ⎠ ⎭
which simplifies to the matrix multiplication operation, where 𝟘 represents the 0 matrix, which simplifies to the matrix multiplication operation, where 𝟘 represents the 0 matrix,
⎛ 1 0 0 ⎞ ⎛ 2 0 0 ⎞ ⎛ 0 0 0 ⎞ ħ ⎛ 1 0 0 ⎞ ħ ⎛ 2 0 0 ⎞ ħ ⎛ 0 0 0 ⎞
ħ ⎜ 0 0 0 ⎟ ħ ⎜ 0 1 0 ⎟ ħ ⎜ 0 -1 0 ⎟ = 𝟘. ⎜ 0 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 -1 0 ⎟ = 𝟘.
⎝ 0 0 -1 ⎠ ⎝ 0 0 0 ⎠ ⎝ 0 0 0 ⎠ ⎝ 0 0 -1 ⎠ ⎝ 0 0 0 ⎠ ⎝ 0 0 0 ⎠
The multiplication operation apparently returns 𝟘 because the third factor will nullify any terms besides center terms, and the first factor will nullify any center terms. The multiplication operation apparently returns 𝟘 because the third factor will nullify any terms besides center terms, and the first factor will nullify any center terms.
Similarly, S𝓍 (S𝓍 + ħ)(S𝓍 - ħ) ≐ Similarly, S𝓍 (S𝓍 + ħ)(S𝓍 - ħ) ≐
⎛ 0 1 0 ⎞ ⎛ 1 1 0 ⎞ ⎛ -1 1 0 ⎞ ⎛ 1 1 1 ⎞ ⎛ -1 1 0 ⎞ ⎛ 0 1 0 ⎞ ͟ħ͟ ⎛ 0 1 0 ⎞ ͟ħ͟ ⎛ √2 1 0 ⎞ ͟ħ͟ ⎛ -√2 1 0 ⎞
ħ³ ⎜ 1 0 1 ⎟ ⎜ 1 1 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 2 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 0 1 ⎟. √2 ⎜ 1 0 1 ⎟ √2 ⎜ 1 √2 1 ⎟ √2 ⎜ 1 -√2 1 ⎟.
⎝ 0 1 0 ⎠ ⎝ 0 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 1 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 0 1 0 ⎠ ⎝ 0 1 0 ⎠ ⎝ 0 1 √2 ⎠ ⎝ 0 1 -√2 ⎠
This expression results in the non-zero matrix Performing the multiplication operation on the last two matrices returns the expression
⎛ 0 1 0 ⎞ ͟ħ͟³͟ ⎛ 0 1 0 ⎞ ⎛ -1 0 1 ⎞
ħ³ ⎜ 1 0 1 ⎟. 2√2 ⎜ 1 0 1 ⎟ ⎜ 0 0 0 ⎟ = 𝟘.
⎝ 0 1 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.

472
solutions/exam1/prob1.ps
View File

@ -1,5 +1,5 @@
%!PS-Adobe-3.0 %!PS-Adobe-3.0
%%Title: Otho Ulrich: Exam 1, #1 %%Title: Otho Ulrich: Exam 1 #1
%%Creator: paps version 0.6.7 by Dov Grobgeld %%Creator: paps version 0.6.7 by Dov Grobgeld
%%Pages: (atend) %%Pages: (atend)
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@ -353,37 +353,6 @@ end_ol
end_ol end_ol
} def } def
/JAA { start_ol /JAA { 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
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4200 1296 4476 2034 x
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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
/KAA { start_ol
2664 6264 m 2664 6264 m
4608 6264 l 4608 6264 l
4608 4104 l 4608 4104 l
@ -395,6 +364,37 @@ end_ol
2664 0 l 2664 0 l
2664 2160 l 2664 2160 l
7200 fwd_x 7200 fwd_x
end_ol
} def
/KAA { 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 5596 991 x
4714 -144 2948 -144 x
2539 -144 2103 -51 x
1668 39 1224 216 x
3465 4392 m
4018 4392 4313 4808 x
4608 5224 4608 6015 x
4608 6799 4313 7215 x
4018 7632 3465 7632 x
2904 7632 2604 7215 x
2304 6799 2304 6015 x
2304 5230 2604 4810 x
2904 4392 3465 4392 x
7200 fwd_x
end_ol end_ol
} def } def
/LAA { start_ol /LAA { start_ol
@ -432,33 +432,33 @@ end_ol
end_ol end_ol
} def } def
/MAA { start_ol /MAA { start_ol
1224 216 m 2880 4388 m
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1680 1505 2100 1364 x 3296 5112 3596 5112 x
2520 1224 2908 1224 x 3903 5112 4111 4903 x
3828 1224 4311 1847 x 4320 4695 4320 4388 x
4794 2470 4824 3710 x 4320 4088 4111 3880 x
4549 3331 4122 3141 x 3903 3672 3596 3672 x
3695 2952 3110 2952 x 3296 3672 3088 3876 x
1935 2952 1291 3703 x 2880 4082 2880 4388 x
648 4456 648 5836 x 3596 7488 m
648 7311 1371 8119 x 2999 7488 2723 6750 x
2095 8928 3420 8928 x 2448 6012 2448 4388 x
5026 8928 5753 7857 x 2448 2772 2723 2034 x
6480 6786 6480 4397 x 2999 1296 3596 1296 x
6480 2127 5596 991 x 4200 1296 4476 2034 x
4714 -144 2948 -144 x 4752 2772 4752 4388 x
2539 -144 2103 -51 x 4752 6012 4476 6750 x
1668 39 1224 216 x 4200 7488 3596 7488 x
3465 4392 m 720 4388 m
4018 4392 4313 4808 x 720 6661 1437 7794 x
4608 5224 4608 6015 x 2155 8928 3596 8928 x
4608 6799 4313 7215 x 5044 8928 5762 7797 x
4018 7632 3465 7632 x 6480 6666 6480 4388 x
2904 7632 2604 7215 x 6480 2116 5762 985 x
2304 6799 2304 6015 x 5044 -144 3596 -144 x
2304 5230 2604 4810 x 2155 -144 1437 988 x
2904 4392 3465 4392 x 720 2122 720 4388 x
7200 fwd_x 7200 fwd_x
end_ol end_ol
} def } def
@ -782,17 +782,6 @@ end_ol
end_ol end_ol
} def } def
/bAA { start_ol /bAA { start_ol
2736 2160 m
4680 2160 l
4680 578 l
3456 -1656 l
2160 -1656 l
2736 578 l
2736 2160 l
7200 fwd_x
end_ol
} def
/cAA { start_ol
4136 8640 m 4136 8640 m
3579 6408 l 3579 6408 l
4771 6408 l 4771 6408 l
@ -830,7 +819,7 @@ end_ol
7200 fwd_x 7200 fwd_x
end_ol end_ol
} def } def
/dAA { start_ol /cAA { start_ol
2664 7344 m 2664 7344 m
2664 4752 l 2664 4752 l
3369 4752 l 3369 4752 l
@ -852,7 +841,7 @@ end_ol
7200 fwd_x 7200 fwd_x
end_ol end_ol
} def } def
/eAA { start_ol /dAA { start_ol
4680 3351 m 4680 3351 m
4680 4214 4351 4735 x 4680 4214 4351 4735 x
4023 5256 3486 5256 x 4023 5256 3486 5256 x
@ -887,7 +876,7 @@ end_ol
7200 fwd_x 7200 fwd_x
end_ol end_ol
} def } def
/fAA { start_ol /eAA { start_ol
3218 5205 m 3218 5205 m
3218 410 l 3218 410 l
4596 410 l 4596 410 l
@ -908,7 +897,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/gAA { start_ol /fAA { start_ol
5171 5544 m 5171 5544 m
5151 5544 4924 5562 x 5151 5544 4924 5562 x
4698 5580 4377 5598 x 4698 5580 4377 5598 x
@ -942,7 +931,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/iAA { start_ol /hAA { start_ol
4608 1542 m 4608 1542 m
4608 1960 4344 2219 x 4608 1960 4344 2219 x
4082 2479 3693 2563 x 4082 2479 3693 2563 x
@ -985,7 +974,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/jAA { start_ol /iAA { start_ol
2724 2941 m 2724 2941 m
2724 2710 2459 2378 x 2724 2710 2459 2378 x
2194 2046 2194 2036 x 2194 2046 2194 2036 x
@ -1037,7 +1026,7 @@ end_ol
4320 fwd_x 4320 fwd_x
end_ol end_ol
} def } def
/lAA { start_ol /kAA { start_ol
1249 3812 m 1249 3812 m
1249 4023 1926 4171 x 1249 4023 1926 4171 x
2603 4320 2981 4320 x 2603 4320 2981 4320 x
@ -1074,7 +1063,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/mAA { start_ol /lAA { start_ol
3207 3909 m 3207 3909 m
3029 3909 2866 3873 x 3029 3909 2866 3873 x
2703 3838 2560 3763 x 2703 3838 2560 3763 x
@ -1115,7 +1104,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/nAA { start_ol /mAA { start_ol
5018 6026 m 5018 6026 m
5018 410 l 5018 410 l
5546 410 l 5546 410 l
@ -1146,7 +1135,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/oAA { start_ol /nAA { start_ol
4097 3450 m 4097 3450 m
4317 3450 4456 3313 x 4317 3450 4456 3313 x
4594 3175 4594 2953 x 4594 3175 4594 2953 x
@ -1194,7 +1183,7 @@ end_ol
4392 fwd_x 4392 fwd_x
end_ol end_ol
} def } def
/pAA { start_ol /oAA { start_ol
5221 3444 m 5221 3444 m
5170 3444 4910 3658 x 5170 3444 4910 3658 x
4650 3873 4399 3873 x 4650 3873 4399 3873 x
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5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/qAA { start_ol /pAA { start_ol
5204 2016 m 5204 2016 m
1040 2016 l 1040 2016 l
1150 1221 1701 743 x 1150 1221 1701 743 x
@ -1250,7 +1239,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/rAA { start_ol /qAA { start_ol
4536 1179 m 4536 1179 m
4536 1545 4189 1743 x 4536 1545 4189 1743 x
3843 1942 3362 2017 x 3843 1942 3362 2017 x
@ -1292,7 +1281,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
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@ -1327,7 +1316,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/tAA { start_ol /sAA { start_ol
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4811 410 l 4811 410 l
@ -1350,7 +1339,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/uAA { start_ol /tAA { start_ol
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5976 fwd_x 5976 fwd_x
end_ol end_ol
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/vAA { start_ol /uAA { start_ol
3218 6142 m 3218 6142 m
3218 410 l 3218 410 l
4605 410 l 4605 410 l
@ -1382,7 +1371,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/wAA { start_ol /vAA { start_ol
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3979 4320 4642 3665 x 3979 4320 4642 3665 x
5306 3011 5306 2063 x 5306 3011 5306 2063 x
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5976 fwd_x 5976 fwd_x
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} def } def
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@ -1434,7 +1423,7 @@ end_ol
5976 fwd_x 5976 fwd_x
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} def } def
/yAA { start_ol /xAA { start_ol
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3702 3909 3145 3909 x 3702 3909 3145 3909 x
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5976 fwd_x 5976 fwd_x
end_ol end_ol
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/zAA { start_ol /yAA { start_ol
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5976 fwd_x 5976 fwd_x
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} def } def
/ABA { start_ol /zAA { start_ol
1418 6026 m 1418 6026 m
1418 3288 l 1418 3288 l
2161 4320 3250 4320 x 2161 4320 3250 4320 x
@ -1523,7 +1512,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/BBA { start_ol /ABA { start_ol
2089 1462 m 2089 1462 m
3420 1462 l 3420 1462 l
1968 -1190 l 1968 -1190 l
@ -1535,7 +1524,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/CBA { start_ol /BBA { start_ol
4413 0 m 4413 0 m
3913 0 l 3913 0 l
3006 2594 l 3006 2594 l
@ -1565,7 +1554,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/DBA { start_ol /CBA { start_ol
4392 0 m 4392 0 m
4392 661 l 4392 661 l
3547 -144 2560 -144 x 3547 -144 2560 -144 x
@ -1593,7 +1582,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/EBA { start_ol /DBA { start_ol
3218 6026 m 3218 6026 m
3218 410 l 3218 410 l
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@ -1611,7 +1600,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/FBA { start_ol /EBA { start_ol
2952 2432 m 2952 2432 m
2952 3172 3252 4057 x 2952 3172 3252 4057 x
3553 4944 3889 5494 x 3553 4944 3889 5494 x
@ -1632,7 +1621,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/GBA { start_ol /FBA { start_ol
3218 2592 m 3218 2592 m
3218 623 l 3218 623 l
3218 360 3018 360 x 3218 360 3018 360 x
@ -1653,7 +1642,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/HBA { start_ol /GBA { start_ol
4392 2886 m 4392 2886 m
4392 3337 4046 3623 x 4392 3337 4046 3623 x
3702 3909 3145 3909 x 3702 3909 3145 3909 x
@ -1700,7 +1689,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/IBA { start_ol /HBA { start_ol
1512 -1040 m 1512 -1040 m
1512 -930 1668 -634 x 1512 -930 1668 -634 x
1824 -339 2016 14 x 1824 -339 2016 14 x
@ -1721,7 +1710,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/KBA { start_ol /JBA { start_ol
3362 0 m 3362 0 m
2661 0 l 2661 0 l
981 3765 l 981 3765 l
@ -1746,7 +1735,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/LBA { start_ol /KBA { start_ol
1368 5278 m 1368 5278 m
1465 5308 1769 5454 x 1465 5308 1769 5454 x
2074 5600 2383 5680 x 2074 5600 2383 5680 x
@ -1782,7 +1771,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/MBA { start_ol /LBA { start_ol
3218 410 m 3218 410 m
4249 410 l 4249 410 l
4514 410 4514 210 x 4514 410 4514 210 x
@ -1808,7 +1797,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/NBA { start_ol /MBA { start_ol
4392 3341 m 4392 3341 m
4392 4176 l 4392 4176 l
5328 4176 l 5328 4176 l
@ -1843,7 +1832,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/OBA { start_ol /NBA { start_ol
1850 1802 m 1850 1802 m
1850 0 l 1850 0 l
914 0 l 914 0 l
@ -1878,7 +1867,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/PBA { start_ol /OBA { start_ol
1130 4176 m 1130 4176 m
1130 3655 l 1130 3655 l
1377 4017 1613 4168 x 1377 4017 1613 4168 x
@ -1921,7 +1910,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/QBA { start_ol /PBA { start_ol
3292 2192 m 3292 2192 m
5134 410 l 5134 410 l
5224 410 l 5224 410 l
@ -1962,7 +1951,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/RBA { start_ol /QBA { start_ol
2822 -6 m 2822 -6 m
940 3765 l 940 3765 l
780 3765 l 780 3765 l
@ -1993,7 +1982,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/SBA { start_ol /RBA { start_ol
2922 1166 m 2922 1166 m
3022 1166 l 3022 1166 l
3313 1166 3513 981 x 3313 1166 3513 981 x
@ -2019,7 +2008,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/TBA { start_ol /SBA { start_ol
3020 5105 m 3020 5105 m
3231 5105 3376 4954 x 3231 5105 3376 4954 x
3521 4804 3521 4594 x 3521 4804 3521 4594 x
@ -2046,7 +2035,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/UBA { start_ol /TBA { start_ol
5133 6865 m 5133 6865 m
3794 5034 3794 2671 x 3794 5034 3794 2671 x
3794 -2001 l 3794 -2001 l
@ -2060,7 +2049,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/VBA { start_ol /UBA { start_ol
4874 3506 m 4874 3506 m
4874 2509 l 4874 2509 l
4874 1332 4356 594 x 4874 1332 4356 594 x
@ -2086,7 +2075,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/WBA { start_ol /VBA { start_ol
842 6836 m 842 6836 m
813 6877 813 6957 x 813 6877 813 6957 x
813 7036 877 7096 x 813 7036 877 7096 x
@ -2102,7 +2091,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/XBA { start_ol /WBA { start_ol
3384 8006 m 3384 8006 m
3794 8006 l 3794 8006 l
3794 -2001 l 3794 -2001 l
@ -2111,7 +2100,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/YBA { start_ol /XBA { start_ol
2642 -2001 m 2642 -2001 m
2232 -2001 l 2232 -2001 l
2232 8006 l 2232 8006 l
@ -2120,7 +2109,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/ZBA { start_ol /YBA { start_ol
5133 -860 m 5133 -860 m
5162 -901 5162 -981 x 5162 -901 5162 -981 x
5162 -1060 5097 -1120 x 5162 -1060 5097 -1120 x
@ -2136,7 +2125,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/aBA { start_ol /ZBA { start_ol
842 -889 m 842 -889 m
2181 941 2181 3304 x 2181 941 2181 3304 x
2181 7977 l 2181 7977 l
@ -2148,9 +2137,65 @@ end_ol
813 -1090 813 -1009 x 813 -1090 813 -1009 x
813 -929 842 -889 x 813 -929 842 -889 x
5976 fwd_x 5976 fwd_x
end_ol
} def
/aBA { start_ol
2981 -651 m
2981 -1152 l
-3024 -1152 l
-3024 -651 l
2981 -651 l
0 fwd_x
end_ol end_ol
} def } def
/bBA { start_ol /bBA { start_ol
1271 3146 m
1821 995 l
3822 6984 l
5784 6984 l
6005 6984 6005 6804 x
6005 6624 5784 6624 x
4113 6624 l
1791 0 l
950 2736 l
320 2736 l
100 2736 100 2946 x
100 3146 320 3146 x
1271 3146 l
5976 fwd_x
end_ol
} def
/cBA { start_ol
1008 4692 m
1008 5142 1562 5656 x
2117 6170 2900 6170 x
3639 6170 4200 5650 x
4763 5132 4763 4452 x
4763 4003 4494 3609 x
4225 3215 3303 2367 x
1202 441 l
1202 410 l
4392 410 l
4392 770 l
4392 1040 4602 1040 x
4802 1040 4802 770 x
4802 0 l
864 0 l
864 600 l
3222 2815 l
3921 3504 4136 3813 x
4351 4123 4351 4462 x
4351 4981 3915 5370 x
3478 5760 2900 5760 x
2382 5760 1960 5460 x
1540 5161 1418 4711 x
1356 4512 1202 4512 x
1131 4512 1069 4567 x
1008 4622 1008 4692 x
5976 fwd_x
end_ol
} def
/dBA { start_ol
2922 1166 m 2922 1166 m
3022 1166 l 3022 1166 l
3313 1166 3513 981 x 3313 1166 3513 981 x
@ -2165,7 +2210,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/cBA { start_ol /eBA { start_ol
5018 5205 m 5018 5205 m
5018 1854 l 5018 1854 l
5018 1009 4439 432 x 5018 1009 4439 432 x
@ -2196,7 +2241,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/dBA { start_ol /fBA { start_ol
5330 873 m 5330 873 m
5330 785 5160 623 x 5330 785 5160 623 x
4990 462 4718 285 x 4990 462 4718 285 x
@ -2226,7 +2271,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/eBA { start_ol /gBA { start_ol
1634 2736 m 1634 2736 m
1634 410 l 1634 410 l
3011 410 l 3011 410 l
@ -2259,7 +2304,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/fBA { start_ol /hBA { start_ol
4749 6624 m 4749 6624 m
4073 6624 3646 6294 x 4073 6624 3646 6294 x
3218 5965 3218 5377 x 3218 5965 3218 5377 x
@ -2273,7 +2318,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/gBA { start_ol /iBA { start_ol
1226 6573 m 1226 6573 m
957 6573 957 6783 x 957 6573 957 6783 x
957 6984 1227 6984 x 957 6984 1227 6984 x
@ -2287,7 +2332,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/hBA { start_ol /jBA { start_ol
3218 8006 m 3218 8006 m
3218 -2001 l 3218 -2001 l
2808 -2001 l 2808 -2001 l
@ -2296,7 +2341,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/iBA { start_ol /kBA { start_ol
4749 -597 m 4749 -597 m
5018 -597 5018 -807 x 5018 -597 5018 -807 x
5018 -1008 4748 -1008 x 5018 -1008 4748 -1008 x
@ -2310,7 +2355,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/jBA { start_ol /lBA { start_ol
1226 -648 m 1226 -648 m
1902 -648 2329 -318 x 1902 -648 2329 -318 x
2757 10 2757 598 x 2757 10 2757 598 x
@ -2324,7 +2369,7 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/kBA { start_ol /mBA { start_ol
1309 6457 m 1309 6457 m
1955 7428 3181 7428 x 1955 7428 3181 7428 x
4407 7428 5058 6457 x 4407 7428 5058 6457 x
@ -2352,37 +2397,7 @@ end_ol
6336 fwd_x 6336 fwd_x
end_ol end_ol
} def } def
/mBA { start_ol /oBA { start_ol
1008 4692 m
1008 5142 1562 5656 x
2117 6170 2900 6170 x
3639 6170 4200 5650 x
4763 5132 4763 4452 x
4763 4003 4494 3609 x
4225 3215 3303 2367 x
1202 441 l
1202 410 l
4392 410 l
4392 770 l
4392 1040 4602 1040 x
4802 1040 4802 770 x
4802 0 l
864 0 l
864 600 l
3222 2815 l
3921 3504 4136 3813 x
4351 4123 4351 4462 x
4351 4981 3915 5370 x
3478 5760 2900 5760 x
2382 5760 1960 5460 x
1540 5161 1418 4711 x
1356 4512 1202 4512 x
1131 4512 1069 4567 x
1008 4622 1008 4692 x
5976 fwd_x
end_ol
} def
/nBA { start_ol
5224 3312 m 5224 3312 m
780 3312 l 780 3312 l
510 3312 510 3522 x 510 3312 510 3522 x
@ -2400,7 +2415,38 @@ end_ol
5976 fwd_x 5976 fwd_x
end_ol end_ol
} def } def
/oBA { start_ol /pBA { start_ol
1634 2325 m
1634 410 l
3011 410 l
3290 410 3290 210 x
3290 0 3012 0 x
699 0 l
432 0 432 210 x
432 410 696 410 x
1224 410 l
1224 5205 l
696 5205 l
432 5205 432 5415 x
432 5616 703 5616 x
3117 5616 l
3921 5616 4469 5143 x
5018 4672 5018 3980 x
5018 3289 4420 2806 x
3822 2325 2962 2325 x
1634 2325 l
1634 2736 m
2994 2736 l
3654 2736 4131 3101 x
4608 3467 4608 3980 x
4608 4484 4176 4844 x
3745 5205 3146 5205 x
1634 5205 l
1634 2736 l
5976 fwd_x
end_ol
} def
/qBA { start_ol
3013 6984 m 3013 6984 m
2810 6984 2655 6925 x 2810 6984 2655 6925 x
2499 6867 2436 6804 x 2499 6867 2436 6804 x
@ -2434,6 +2480,40 @@ end_ol
3672 6666 3483 6825 x 3672 6666 3483 6825 x
3294 6984 3013 6984 x 3294 6984 3013 6984 x
5976 fwd_x 5976 fwd_x
end_ol
} def
/rBA { start_ol
4608 3351 m
4608 4176 l
5544 4176 l
5810 4176 5810 3978 x
5810 3765 5546 3765 x
5018 3765 l
5018 -1461 l
5546 -1461 l
5810 -1461 5810 -1661 x
5810 -1872 5545 -1872 x
3649 -1872 l
3384 -1872 3384 -1661 x
3384 -1461 3648 -1461 x
4608 -1461 l
4608 1038 l
3942 72 2818 72 x
1902 72 1274 686 x
648 1300 648 2199 x
648 3098 1274 3709 x
1902 4320 2828 4320 x
3931 4320 4608 3351 x
2828 3909 m
2082 3909 1570 3413 x
1058 2917 1058 2199 x
1058 1482 1570 982 x
2082 482 2828 482 x
3564 482 4086 977 x
4608 1472 4608 2179 x
4608 2918 4096 3413 x
3584 3909 2828 3909 x
5976 fwd_x
end_ol end_ol
} def } def
end end end end
@ -2441,50 +2521,60 @@ end end
%%EndPrologue %%EndPrologue
%%Page: 1 1 %%Page: 1 1
paps_bop paps_bop
(+ 36000 807000AAABAACAA+ 64800 807000EAABAAFAA+ 93600 807000GAAHAA+ 115200 807000IAAJAAKAAGAALAAKAAHAAMAA+ 180000 807000GAAJAAIAANAA)paps_exec (+ 36000 807000AAABAACAA+ 64800 807000EAABAAFAA+ 93600 807000GAAHAA+ 115200 807000IAAHAAJAAIAAKAAJAALAAMAA+ 180000 807000GAAMAAIAANAA)paps_exec
(>2950OAAPAAQAARAA+ 250700 807000SAATAAUAAVAAWAAQAAKAA+ 308300 807000XAAYAAZAAaAA+ 344300 807000IAAbAA+ 365900 807000cAAIAA)paps_exec (>4750OAAPAAQAARAA+ 254300 807000SAATAAUAAVAAWAAQAAJAA+ 311900 807000XAAYAAZAAaAA+ 347900 807000IAA+ 362300 807000bAAIAA)paps_exec
(* 515800dAAZAAeAABAA+ 551800 807000IAA)paps_exec (* 515800cAAZAAdAABAA+ 551800 807000IAA)paps_exec
36 797.000000 moveto 559 797.000000 lineto 0 setlinewidth stroke 36 797.000000 moveto 559 797.000000 lineto 0 setlinewidth stroke
(+ 36000 779984fAAgAA+ 53928 779984iAAjAA+ 66744 779984lAAmAAnAA+ 90648 779984iAAoAA+ 103536 779984lAApAAqAA+ 127440 779984rAAsAAtAAmAAuAAvAA+ 169272 779984wAAsAAqAApAAlAAxAAwAApAArAA+ 229032 779984tAAmAA+ 246960 779984xAAyAAqAA+ 270864 779984zAA+ 282816 779984ABAlAArAAtAArAABBA+ 324648 779984CBAyAAlAAxAA+ 354528 779984lAApAAqAA+ 378432 779984xAAyAAqAA+ 402336 779984pAAqAArAADBAEBAxAArAA+ 450144 779984tAAgAA+ 468072 779984iAAjAA+ 480888 779984FBAiAAjAA+ 499680 779984GBA+ 511632 779984HBAIBAFBAiAAjAA+ 542376 779984uAA)paps_exec (+ 36000 779984eAAfAA+ 53928 779984hAAiAA+ 66744 779984kAAlAAmAA+ 90648 779984hAAnAA+ 103536 779984kAAoAApAA+ 127440 779984qAArAAsAAlAAtAAuAA+ 169272 779984vAArAApAAoAAkAAwAAvAAoAAqAA+ 229032 779984sAAlAA+ 246960 779984wAAxAApAA+ 270864 779984yAA+ 282816 779984zAAkAAqAAsAAqAAABA+ 324648 779984BBAxAAkAAwAA+ 354528 779984kAAoAApAA+ 378432 779984wAAxAApAA+ 402336 779984oAApAAqAACBADBAwAAqAA+ 450144 779984sAAfAA+ 468072 779984hAAiAA+ 480888 779984EBAhAAiAA+ 499680 779984FBA+ 511632 779984GBAHBAEBAhAAiAA+ 542376 779984tAA)paps_exec
(+ 36000 768968HBAIBA+ 53928 768968lAAmAAnAA+ 77832 768968iAAoAA+ 90720 768968FBAiAAoAA+ 109584 768968GBA+ 121536 768968HBAIBAFBAiAAoAA+ 152352 768968uAA+ 164304 768968HBAIBA+ 182232 768968lAApAAqAA+ 206136 768968qAAKBAlAAEBADBAlAAxAAqAAnAALBA)paps_exec (+ 36000 768968GBAHBA+ 53928 768968kAAlAAmAA+ 77832 768968hAAnAA+ 90720 768968EBAhAAnAA+ 109584 768968FBA+ 121536 768968GBAHBAEBAhAAnAA+ 152352 768968tAA+ 164304 768968GBAHBA+ 182232 768968kAAoAApAA+ 206136 768968pAAJBAkAADBACBAkAAwAApAAmAAKBA)paps_exec
()paps_exec ()paps_exec
(+ 36000 748808MBAyAAqAA+ 59904 748808rAAsAAtAAmAA+ 89784 748808qAAtAANBAqAAmAArAAxAAlAAxAAqAArAA+ 161496 748808lAAmAAnAA+ 185400 748808qAAtAANBAqAAmAAKBAlAAEBADBAqAArAA+ 257112 748808lAApAAqAA+ 281016 748808OBAmAAwAACBAmAA+ 316872 748808gAApAAwAAPBA+ 346752 748808qAAQBAsAAqAApAAtAAPBAqAAmAAxAA+ 412488 748808gAAwAApAA+ 436392 748808lAA+ 448344 748808rAAsAAtAAmAAuAAvAA+ 490176 748808rAARBArAAxAAqAAPBABBA)paps_exec (+ 36000 748808LBAxAApAA+ 59904 748808qAArAAsAAlAA+ 89784 748808pAAsAAMBApAAlAAqAAwAAkAAwAApAAqAA+ 161496 748808kAAlAAmAA+ 185400 748808pAAsAAMBApAAlAAJBAkAADBACBApAAqAA+ 257112 748808kAAoAApAA+ 281016 748808NBAlAAvAABBAlAA+ 316872 748808fAAoAAvAAOBA+ 346752 748808pAAPBArAApAAoAAsAAOBApAAlAAwAA+ 412488 748808fAAvAAoAA+ 436392 748808kAA+ 448344 748808qAArAAsAAlAAtAAuAA+ 490176 748808qAAQBAqAAwAApAAOBAABA)paps_exec
(+ 36000 737792lAAmAAnAA+ 59904 737792xAAyAAqAA+ 83808 737792rAAsAAtAAmAAuAAzAA+ 125640 737792lAAmAAnAA+ 149544 737792rAAsAAtAAmAAuAAQBA+ 191376 737792wAAsAAqAApAAlAAxAAwAApAArAA+ 251136 737792tAAmAA+ 269064 737792xAAyAAqAA+ 292968 737792zAA+ 304920 737792ABAlAArAAtAArAABBA+ 346752 737792iAAjAA+ 359568 737792lAAmAAnAA+ 383472 737792iAAoAABBA+ 405792 737792yAAlAAKBAqAA+ 435672 737792xAAyAAqAA+ 459576 737792gAAwAAEBAEBAwAACBAtAAmAANBA)paps_exec (+ 36000 737792kAAlAAmAA+ 59904 737792wAAxAApAA+ 83808 737792qAArAAsAAlAAtAAyAA+ 125640 737792kAAlAAmAA+ 149544 737792qAArAAsAAlAAtAAPBA+ 191376 737792vAArAApAAoAAkAAwAAvAAoAAqAA+ 251136 737792sAAlAA+ 269064 737792wAAxAApAA+ 292968 737792yAA+ 304920 737792zAAkAAqAAsAAqAAABA+ 346752 737792hAAiAA+ 359568 737792kAAlAAmAA+ 383472 737792hAAnAAABA+ 405792 737792xAAkAAJBApAA+ 435672 737792wAAxAApAA+ 459576 737792fAAvAADBADBAvAABBAsAAlAAMBA)paps_exec
(+ 36000 727712PBAlAAxAApAAtAAQBA+ 77832 727712pAAqAAsAApAAqAArAAqAAmAAxAAlAAxAAtAAwAAmAArAASBA)paps_exec (+ 36000 727712OBAkAAwAAoAAsAAPBA+ 77832 727712oAApAArAAoAApAAqAApAAlAAwAAkAAwAAsAAvAAlAAqAARBA)paps_exec
()paps_exec ()paps_exec
()paps_exec ()paps_exec
(+ 36000 696536iAAjAA+ 48816 696536TBA)paps_exec (+ 36000 696536hAAiAA+ 48816 696536SBA)paps_exec
(+ 83808 686456UBA+ 95760 686456vAA+ 107712 686456VBA+ 125640 686456VBA+ 137592 686456WBA)paps_exec (+ 71856 686456GBA+ 83808 686456TBA+ 95760 686456uAA+ 107712 686456UBA+ 125640 686456UBA+ 137592 686456VBA)paps_exec
(+ 71856 676376HBA+ 83808 676376XBA+ 95760 676376VBA+ 107712 676376VBA+ 125640 676376VBA+ 137592 676376YBA+ 149544 676376lAAmAAnAA)paps_exec (+ 83808 676376WBA+ 95760 676376UBA+ 107712 676376UBA+ 125640 676376UBA+ 137592 676376XBA+ 149544 676376kAAlAAmAA)paps_exec
(+ 83808 666296ZBA+ 95760 666296VBA+ 107712 666296VBA+ 119664 666296uAAvAA+ 137592 666296aBA)paps_exec (+ 83808 666296YBA+ 95760 666296UBA+ 107712 666296UBA+ 119664 666296tAAuAA+ 137592 666296ZBA)paps_exec
(+ 36000 655280iAAoAA+ 48888 655280TBA)paps_exec (+ 36000 655280hAAnAA+ 48888 655280SBA)paps_exec
(+ 83808 645200UBA+ 95760 645200VBA+ 107712 645200vAA+ 119664 645200VBA+ 131616 645200WBA)paps_exec (+ 71856 645200aBAGBAaBA+ 89784 645200TBA+ 101736 645200UBA+ 113688 645200uAA+ 125640 645200UBA+ 137592 645200VBA)paps_exec
(+ 71856 635120HBA+ 83808 635120XBA+ 95760 635120vAA+ 107712 635120VBA+ 119664 635120vAA+ 131616 635120YBAbBA)paps_exec (+ 71856 635120bBAcBA+ 89784 635120WBA+ 101736 635120uAA+ 113688 635120UBA+ 125640 635120uAA+ 137592 635120XBAdBA)paps_exec
(+ 83808 625040ZBA+ 95760 625040VBA+ 107712 625040vAA+ 119664 625040VBA+ 131616 625040aBA)paps_exec (+ 89784 625040YBA+ 101736 625040UBA+ 113688 625040uAA+ 125640 625040UBA+ 137592 625040ZBA)paps_exec
()paps_exec ()paps_exec
(+ 36000 604880cBArAAtAAmAANBA+ 71856 604880xAAyAAqAA+ 95760 604880PBAlAAxAApAAtAAQBA+ 137592 604880pAAqAAsAApAAqAArAAqAAmAAxAAlAAxAAtAAwAAmAArAABBA+ 239184 604880xAAyAAqAA+ 263088 604880qAAQBAsAApAAqAArAArAAtAAwAAmAArAA+ 334800 604880dBAlAAmAA+ 358704 604880ABAqAA+ 376632 604880qAAKBAlAAEBADBAlAAxAAqAAnAAbBA+ 442368 604880eBAwAApAA+ 466272 604880xAAyAAqAA+ 490176 604880rAAsAAtAAmAAuAAzAA)paps_exec (+ 36000 604880eBAqAAsAAlAAMBA+ 71856 604880wAAxAApAA+ 95760 604880OBAkAAwAAoAAsAAPBA+ 137592 604880oAApAArAAoAApAAqAApAAlAAwAAkAAwAAsAAvAAlAAqAAABA+ 239184 604880wAAxAApAA+ 263088 604880pAAPBArAAoAApAAqAAqAAsAAvAAlAAqAA+ 334800 604880fBAkAAlAA+ 358704 604880zAApAA+ 376632 604880pAAJBAkAADBACBAkAAwAApAAmAAdBA+ 442368 604880gBAvAAoAA+ 466272 604880wAAxAApAA+ 490176 604880qAArAAsAAlAAtAAyAA)paps_exec
(+ 36000 593864wAAsAAqAApAAlAAxAAwAApAABBA+ 95760 593864xAAyAAqAA+ 119664 593864qAAQBAsAApAAqAArAArAAtAAwAAmAA+ 185400 593864iAAjAA+ 198216 593864FBAiAAjAA+ 217008 593864GBA+ 228960 593864HBAIBAFBAiAAjAA+ 259704 593864uAA+ 271656 593864HBAIBA+ 289584 593864TBA)paps_exec (+ 36000 593864vAArAApAAoAAkAAwAAvAAoAAABA+ 95760 593864wAAxAApAA+ 119664 593864pAAPBArAAoAApAAqAAqAAsAAvAAlAA+ 185400 593864hAAiAA+ 198216 593864EBAhAAiAA+ 217008 593864FBA+ 228960 593864GBAHBAEBAhAAiAA+ 259704 593864tAA+ 271656 593864GBAHBA+ 289584 593864SBA)paps_exec
()paps_exec ()paps_exec
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()paps_exec
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()paps_exec
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()paps_exec
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paps_eop paps_eop
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%%Pages: 1 %%Pages: 1

87
solutions/exam1/prob3
View File

@ -1,8 +1,89 @@
|a〉 and |b〉 are eigenstates of a Hermitian operator A with eigenvalues a and b, a ≠ b. The Hamiltonian operator is |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. Ĥ = |a〉 δ 〈a| + |b〉 δ 〈b|, with δ a real number.
a) The eigenstates of the Hamiltonian can be determined by a) The eigenstates of the Hamiltonian can be determined by diagonalizing the Hamiltonian operator's matrix representation. In general,
🔋 Ĥ ≐
⎛ 〈a|Ĥ|a〉 〈a|Ĥ|b〉 ⎞
⎝ 〈b|Ĥ|a〉 〈b|Ĥ|b〉 ⎠.
Calculating the individual components:
〈a|Ĥ|a〉 = 〈a|(|a〉 δ 〈a| + |b〉 δ 〈b|)|a〉 =
〈a|Ĥ|a〉 = 〈a|a〉 δ 〈a|a〉 + 〈a|b〉 δ 〈b|a〉 =
〈a|Ĥ|a〉 = δ(1 + 〈a|b〉〈b|a〉),
and because H is a hermitian operator, 〈a|b〉 = 〈b|a〉, so
〈a|Ĥ|a〉 = δ(1 + |〈a|b〉|²);
〈a|Ĥ|b〉 = 〈a|a〉 δ 〈a|b〉 + 〈a|b〉 δ 〈b|b〉 =
〈a|Ĥ|b〉 = δ (〈a|b〉 + 〈a|b〉) = δ 2〈a|b〉;
because of the hermition property,
〈b|Ĥ|a〉 = 〈a|Ĥ|b〉 = δ 2〈a|b〉;
finally,
〈b|Ĥ|b〉 = 〈b|(|a〉 δ 〈a| + |b〉 δ 〈b|)|b〉 =
〈b|Ĥ|b〉 = 〈b|a〉 δ 〈a|b〉 + 〈b|b〉 δ 〈b|b〉 =
〈b|Ĥ|b〉 = δ(〈b|a〉〈a|b〉 + 1),
〈b|Ĥ|b〉 = δ(1 + |〈a|b〉|²).
So, the Hamiltonian operator Ĥ ≐
δ ⎛ 1 + |〈a|b〉|² 2〈a|b〉 ⎞
⎝ 2〈a|b〉 1 + |〈a|b〉|² ⎠.
The eigenstates, which I will call |1〉 and |2〉 can be obtained by diagonalizing the Hamiltonian matrix. The first eigenvalue equations are
Ĥ|1〉 = E₁|1〉 and Ĥ|2〉 = E₂|2〉, with the eigenstates represented by the vector matrices, respectively,
⎛α₁⎞ ⎛α₂⎞
⎝β₁⎠ and ⎝β₂⎠.
δ ⎛ 1 + |〈a|b〉|² 2〈a|b〉 ⎞ ⎛α₁⎞ = E₁ ⎛α₁⎞
⎝ 2〈a|b〉 1 + |〈a|b〉|² ⎠ ⎝β₁⎠ ⎝β₁⎠.
This gives the equation α₁ + α₁|〈a|b〉|² + 2β₁〈a|b〉 = E₁α₁, and therefore the ratio between α₁ and β₁,
͟β͟₁͟ = ͟E͟₁͟ ͟-͟ ͟1͟ ͟-͟ ͟|͟〈͟a͟|͟b͟〉͟|͟²͟, or
α₁ 2〈a|b〉
β₁ = ͟α͟₁͟(͟E͟₁͟ ͟-͟ ͟1͟ ͟-͟ ͟|͟〈͟a͟|͟b͟〉͟|͟²͟)͟
2〈a|b〉
Using the normalization condition, the values of each constant can be obtained. Plugging the value for α₁ into the equation reveals a quadratic equation.
|α₁|² + |β₁|² = 1, so
|α₁|² + | ͟α͟₁͟(͟E͟₁͟ ͟-͟ ͟1͟ ͟-͟ ͟|͟〈͟a͟|͟b͟〉͟|͟²͟)͟ |² = 1.
| 2〈a|b〉 |
α₁(1 - E₁) + 2β₁〈a|b〉 + α₁|〈a|b〉|² = 0 and
_________
α₁ = ±√1 - |β₁|², so
_________ _________
±√̅1 - |β₁|² (1 - E₁) + 2β₁〈a|b〉 + ±√1 - |β₁|² |〈a|b〉|² = 0.
The quadratic formula therefore says that

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Two operators' matrix representations are known in the |1〉, |2〉, |3〉 basis, where and b are real numbers:
A ≐ B ≐
⎛ a 0 0 ⎞ ⎛ b 0 0 ⎞
⎜ 0 -a 0 ⎟ ⎜ 0 0 -ιb ⎟
⎝ 0 0 -a ⎠ and ⎝ 0 ιb 0 ⎠.
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)(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.
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
[Â,B̂] = Â B̂ - B̂ Â.
For the given operators, then, the commutator is
⎛ a 0 0 ⎞ ⎛ b 0 0 ⎞ ⎛ b 0 0 ⎞ ⎛ a 0 0 ⎞
⎜ 0 -a 0 ⎟ ⎜ 0 0 -ιb ⎟ - ⎜ 0 0 -ιb ⎟ ⎜ 0 -a 0 ⎟
⎝ 0 0 -a ⎠ ⎝ 0 ιb 0 ⎠ ⎝ 0 ιb 0 ⎠ ⎝ 0 0 -a ⎠,
which reduces to
⎛ ab 0 0 ⎞ ⎛ ab 0 0 ⎞
⎜ 0 0 ιab ⎟ - ⎜ 0 0 ιab ⎟ = 0.
⎝ 0 -ιab 0 ⎠ ⎝ 0 -ιab 0 ⎠
(b) Therefore, these operators commute.
Since the operators commute, they share common eigenstates. Therefore, if the eigenstates for one operator can be determined, they are determined for both operators. Since the problem states that a new set of orthonormal kets need be determined, and the given set are the eigenstates related to the operator A, then the eigenstates of the operator B should be determined.
The eigenvalues are already known (λ = b,b,-b.), and using the eigenvalue equations, the eigenstates can be determined. The eigenvalue equation
⎛ b 0 0 ⎞ ⎛ α₁ ⎞ ⎛ b α₁ ⎞ ⎛ α₁ ⎞
⎜ 0 0 -ιb ⎟ ⎜ β₁ ⎟ = ⎜ -ι b γ₁ ⎟ = b ⎜ β₁ ⎟
⎝ 0 ιb 0 ⎠ ⎝ γ₁ ⎠ ⎝ ι b β₁ ⎠ ⎝ γ₁ ⎠ reveals
-ι γ₁ = β₁, and
ι b β₁ = γ₁

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