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still working on exam 1
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@ -4,36 +4,38 @@ The spin eigenstates and eigenvalues are known from experiment for a spin-1 syst
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S𝓏 ≐
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⎛ 1 0 0 ⎞
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ħ ⎜ 0 0 0 ⎟ and
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ħ ⎛ 1 0 0 ⎞
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⎜ 0 0 0 ⎟ and
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⎝ 0 0 -1 ⎠
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S𝓍 ≐
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⎛ 0 1 0 ⎞
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ħ ⎜ 1 0 1 ⎟.
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⎝ 0 1 0 ⎠
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͟ħ͟ ⎛ 0 1 0 ⎞
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√2 ⎜ 1 0 1 ⎟.
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⎝ 0 1 0 ⎠
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Using the matrix representations, the expressions can be evaluated. For the spin-z operator, the expression S𝓏 (S𝓏 + ħ)(S𝓏 - ħ) ≐
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⎛ 1 0 0 ⎞ ⎧ ⎛ 1 0 0 ⎞ ⎛ 1 0 0 ⎞ ⎫ ⎧ ⎛ 1 0 0 ⎞ ⎛ 1 0 0 ⎞ ⎫
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ħ ⎜ 0 0 0 ⎟ ⎪ ħ ⎜ 0 0 0 ⎟ + ħ ⎜ 0 1 0 ⎟ ⎪ ⎪ ħ ⎜ 0 0 0 ⎟ - ħ ⎜ 0 1 0 ⎟ ⎪,
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ħ ⎛ 1 0 0 ⎞ ⎧ ħ ⎛ 1 0 0 ⎞ ħ ⎛ 1 0 0 ⎞ ⎫ ⎧ ħ ⎛ 1 0 0 ⎞ ħ ⎛ 1 0 0 ⎞ ⎫
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⎜ 0 0 0 ⎟ ⎪ ⎜ 0 0 0 ⎟ + ⎜ 0 1 0 ⎟ ⎪ ⎪ ⎜ 0 0 0 ⎟ - ⎜ 0 1 0 ⎟ ⎪,
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⎝ 0 0 -1 ⎠ ⎩ ⎝ 0 0 -1 ⎠ ⎝ 0 0 1 ⎠ ⎭ ⎩ ⎝ 0 0 -1 ⎠ ⎝ 0 0 1 ⎠ ⎭
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which simplifies to the matrix multiplication operation, where 𝟘 represents the 0 matrix,
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⎛ 1 0 0 ⎞ ⎛ 2 0 0 ⎞ ⎛ 0 0 0 ⎞
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ħ ⎜ 0 0 0 ⎟ ħ ⎜ 0 1 0 ⎟ ħ ⎜ 0 -1 0 ⎟ = 𝟘.
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ħ ⎛ 1 0 0 ⎞ ħ ⎛ 2 0 0 ⎞ ħ ⎛ 0 0 0 ⎞
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⎜ 0 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 -1 0 ⎟ = 𝟘.
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⎝ 0 0 -1 ⎠ ⎝ 0 0 0 ⎠ ⎝ 0 0 0 ⎠
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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.
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Similarly, S𝓍 (S𝓍 + ħ)(S𝓍 - ħ) ≐
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⎛ 0 1 0 ⎞ ⎛ 1 1 0 ⎞ ⎛ -1 1 0 ⎞ ⎛ 1 1 1 ⎞ ⎛ -1 1 0 ⎞ ⎛ 0 1 0 ⎞
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ħ³ ⎜ 1 0 1 ⎟ ⎜ 1 1 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 2 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 0 1 ⎟.
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⎝ 0 1 0 ⎠ ⎝ 0 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 1 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 0 1 0 ⎠
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͟ħ͟ ⎛ 0 1 0 ⎞ ͟ħ͟ ⎛ √2 1 0 ⎞ ͟ħ͟ ⎛ -√2 1 0 ⎞
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√2 ⎜ 1 0 1 ⎟ √2 ⎜ 1 √2 1 ⎟ √2 ⎜ 1 -√2 1 ⎟.
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⎝ 0 1 0 ⎠ ⎝ 0 1 √2 ⎠ ⎝ 0 1 -√2 ⎠
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This expression results in the non-zero matrix
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Performing the multiplication operation on the last two matrices returns the expression
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⎛ 0 1 0 ⎞
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ħ³ ⎜ 1 0 1 ⎟.
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⎝ 0 1 0 ⎠
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͟ħ͟³͟ ⎛ 0 1 0 ⎞ ⎛ -1 0 1 ⎞
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2√2 ⎜ 1 0 1 ⎟ ⎜ 0 0 0 ⎟ = 𝟘.
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⎝ 0 1 0 ⎠ ⎝ 1 0 -1 ⎠
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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.
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@ -1,5 +1,5 @@
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%!PS-Adobe-3.0
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%%Title: Otho Ulrich: Exam 1, #1
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%%Title: Otho Ulrich: Exam 1 #1
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%%Creator: paps version 0.6.7 by Dov Grobgeld
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%%Pages: (atend)
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%%BoundingBox: 0 0 595 841
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@ -353,37 +353,6 @@ end_ol
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end_ol
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} def
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/JAA { start_ol
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2880 4388 m
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2880 4695 3088 4903 x
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3296 5112 3596 5112 x
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3903 5112 4111 4903 x
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4320 4695 4320 4388 x
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4320 4088 4111 3880 x
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3903 3672 3596 3672 x
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3296 3672 3088 3876 x
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2880 4082 2880 4388 x
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3596 7488 m
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2999 7488 2723 6750 x
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2448 6012 2448 4388 x
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2448 2772 2723 2034 x
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2999 1296 3596 1296 x
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4200 1296 4476 2034 x
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4752 2772 4752 4388 x
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4752 6012 4476 6750 x
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4200 7488 3596 7488 x
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720 4388 m
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720 6661 1437 7794 x
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2155 8928 3596 8928 x
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5044 8928 5762 7797 x
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6480 6666 6480 4388 x
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6480 2116 5762 985 x
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5044 -144 3596 -144 x
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2155 -144 1437 988 x
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720 2122 720 4388 x
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7200 fwd_x
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end_ol
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} def
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/KAA { start_ol
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2664 6264 m
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4608 6264 l
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4608 4104 l
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@ -395,6 +364,37 @@ end_ol
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2664 0 l
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2664 2160 l
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7200 fwd_x
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end_ol
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} def
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/KAA { start_ol
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1224 216 m
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1224 1800 l
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1680 1505 2100 1364 x
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2520 1224 2908 1224 x
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3828 1224 4311 1847 x
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4794 2470 4824 3710 x
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4549 3331 4122 3141 x
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3695 2952 3110 2952 x
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1935 2952 1291 3703 x
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648 4456 648 5836 x
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648 7311 1371 8119 x
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2095 8928 3420 8928 x
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5026 8928 5753 7857 x
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6480 6786 6480 4397 x
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6480 2127 5596 991 x
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4714 -144 2948 -144 x
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2539 -144 2103 -51 x
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1668 39 1224 216 x
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3465 4392 m
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4018 4392 4313 4808 x
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4608 5224 4608 6015 x
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4608 6799 4313 7215 x
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4018 7632 3465 7632 x
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2904 7632 2604 7215 x
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2304 6799 2304 6015 x
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2304 5230 2604 4810 x
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2904 4392 3465 4392 x
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7200 fwd_x
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end_ol
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} def
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/LAA { start_ol
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@ -432,33 +432,33 @@ end_ol
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end_ol
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} def
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/MAA { start_ol
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1224 216 m
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1224 1800 l
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1680 1505 2100 1364 x
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2520 1224 2908 1224 x
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3828 1224 4311 1847 x
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4794 2470 4824 3710 x
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4549 3331 4122 3141 x
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3695 2952 3110 2952 x
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1935 2952 1291 3703 x
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648 4456 648 5836 x
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648 7311 1371 8119 x
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2095 8928 3420 8928 x
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5026 8928 5753 7857 x
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6480 6786 6480 4397 x
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6480 2127 5596 991 x
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4714 -144 2948 -144 x
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2539 -144 2103 -51 x
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1668 39 1224 216 x
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3465 4392 m
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4018 4392 4313 4808 x
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4608 5224 4608 6015 x
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4608 6799 4313 7215 x
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4018 7632 3465 7632 x
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2904 7632 2604 7215 x
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2304 6799 2304 6015 x
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2304 5230 2604 4810 x
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2904 4392 3465 4392 x
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2880 4388 m
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2880 4695 3088 4903 x
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3296 5112 3596 5112 x
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3903 5112 4111 4903 x
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4320 4695 4320 4388 x
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4320 4088 4111 3880 x
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3903 3672 3596 3672 x
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3296 3672 3088 3876 x
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2880 4082 2880 4388 x
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3596 7488 m
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2999 7488 2723 6750 x
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2448 6012 2448 4388 x
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2448 2772 2723 2034 x
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2999 1296 3596 1296 x
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4200 1296 4476 2034 x
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4752 2772 4752 4388 x
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4752 6012 4476 6750 x
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4200 7488 3596 7488 x
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720 4388 m
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720 6661 1437 7794 x
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2155 8928 3596 8928 x
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5044 8928 5762 7797 x
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6480 6666 6480 4388 x
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6480 2116 5762 985 x
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5044 -144 3596 -144 x
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2155 -144 1437 988 x
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720 2122 720 4388 x
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7200 fwd_x
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end_ol
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} def
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@ -782,17 +782,6 @@ end_ol
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end_ol
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} def
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/bAA { start_ol
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2736 2160 m
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4680 2160 l
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4680 578 l
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3456 -1656 l
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2160 -1656 l
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2736 578 l
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2736 2160 l
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7200 fwd_x
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end_ol
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} def
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/cAA { start_ol
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4136 8640 m
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3579 6408 l
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4771 6408 l
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@ -830,7 +819,7 @@ end_ol
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7200 fwd_x
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end_ol
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} def
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/dAA { start_ol
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/cAA { start_ol
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2664 7344 m
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2664 4752 l
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3369 4752 l
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@ -852,7 +841,7 @@ end_ol
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7200 fwd_x
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end_ol
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} def
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/eAA { start_ol
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/dAA { start_ol
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4680 3351 m
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4680 4214 4351 4735 x
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4023 5256 3486 5256 x
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@ -887,7 +876,7 @@ end_ol
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7200 fwd_x
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end_ol
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} def
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/fAA { start_ol
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/eAA { start_ol
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3218 5205 m
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3218 410 l
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4596 410 l
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@ -908,7 +897,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/gAA { start_ol
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/fAA { start_ol
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5171 5544 m
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5151 5544 4924 5562 x
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4698 5580 4377 5598 x
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@ -942,7 +931,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/iAA { start_ol
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/hAA { start_ol
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4608 1542 m
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4608 1960 4344 2219 x
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4082 2479 3693 2563 x
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@ -985,7 +974,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/jAA { start_ol
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/iAA { start_ol
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2724 2941 m
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2724 2710 2459 2378 x
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2194 2046 2194 2036 x
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@ -1037,7 +1026,7 @@ end_ol
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4320 fwd_x
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end_ol
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} def
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/lAA { start_ol
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/kAA { start_ol
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1249 3812 m
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1249 4023 1926 4171 x
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2603 4320 2981 4320 x
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@ -1074,7 +1063,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/mAA { start_ol
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/lAA { start_ol
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3207 3909 m
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3029 3909 2866 3873 x
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2703 3838 2560 3763 x
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@ -1115,7 +1104,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/nAA { start_ol
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/mAA { start_ol
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5018 6026 m
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5018 410 l
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5546 410 l
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@ -1146,7 +1135,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/oAA { start_ol
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/nAA { start_ol
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4097 3450 m
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4317 3450 4456 3313 x
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4594 3175 4594 2953 x
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@ -1194,7 +1183,7 @@ end_ol
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4392 fwd_x
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end_ol
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} def
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/pAA { start_ol
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/oAA { start_ol
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5221 3444 m
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5170 3444 4910 3658 x
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4650 3873 4399 3873 x
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@ -1223,7 +1212,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/qAA { start_ol
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/pAA { start_ol
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5204 2016 m
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1040 2016 l
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1150 1221 1701 743 x
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@ -1250,7 +1239,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/rAA { start_ol
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/qAA { start_ol
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4536 1179 m
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4536 1545 4189 1743 x
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3843 1942 3362 2017 x
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@ -1292,7 +1281,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/sAA { start_ol
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/rAA { start_ol
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1418 4176 m
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1418 3351 l
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1825 3875 2242 4097 x
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@ -1327,7 +1316,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/tAA { start_ol
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/sAA { start_ol
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3218 4176 m
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3218 410 l
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4811 410 l
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@ -1350,7 +1339,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/uAA { start_ol
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/tAA { start_ol
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5014 2592 m
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991 2592 l
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721 2592 721 2802 x
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@ -1361,7 +1350,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/vAA { start_ol
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/uAA { start_ol
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3218 6142 m
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3218 410 l
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4605 410 l
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@ -1382,7 +1371,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/wAA { start_ol
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/vAA { start_ol
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3013 4320 m
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3979 4320 4642 3665 x
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5306 3011 5306 2063 x
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@ -1404,7 +1393,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/xAA { start_ol
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/wAA { start_ol
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1850 4176 m
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4030 4176 l
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4298 4176 4298 3975 x
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@ -1434,7 +1423,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/yAA { start_ol
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/xAA { start_ol
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4392 2886 m
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4392 3337 4046 3623 x
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3702 3909 3145 3909 x
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@ -1471,7 +1460,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/zAA { start_ol
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/yAA { start_ol
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4658 4176 m
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4658 3816 l
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1694 410 l
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@ -1492,7 +1481,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/ABA { start_ol
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/zAA { start_ol
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1418 6026 m
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1418 3288 l
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2161 4320 3250 4320 x
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@ -1523,7 +1512,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/BBA { start_ol
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/ABA { start_ol
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2089 1462 m
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3420 1462 l
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1968 -1190 l
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@ -1535,7 +1524,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/CBA { start_ol
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/BBA { start_ol
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4413 0 m
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3913 0 l
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3006 2594 l
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@ -1565,7 +1554,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/DBA { start_ol
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/CBA { start_ol
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4392 0 m
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4392 661 l
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3547 -144 2560 -144 x
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@ -1593,7 +1582,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/EBA { start_ol
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/DBA { start_ol
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3218 6026 m
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3218 410 l
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4811 410 l
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@ -1611,7 +1600,7 @@ end_ol
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5976 fwd_x
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end_ol
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} def
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/FBA { start_ol
|
||||
/EBA { start_ol
|
||||
2952 2432 m
|
||||
2952 3172 3252 4057 x
|
||||
3553 4944 3889 5494 x
|
||||
@ -1632,7 +1621,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/GBA { start_ol
|
||||
/FBA { start_ol
|
||||
3218 2592 m
|
||||
3218 623 l
|
||||
3218 360 3018 360 x
|
||||
@ -1653,7 +1642,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/HBA { start_ol
|
||||
/GBA { start_ol
|
||||
4392 2886 m
|
||||
4392 3337 4046 3623 x
|
||||
3702 3909 3145 3909 x
|
||||
@ -1700,7 +1689,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/IBA { start_ol
|
||||
/HBA { start_ol
|
||||
1512 -1040 m
|
||||
1512 -930 1668 -634 x
|
||||
1824 -339 2016 14 x
|
||||
@ -1721,7 +1710,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/KBA { start_ol
|
||||
/JBA { start_ol
|
||||
3362 0 m
|
||||
2661 0 l
|
||||
981 3765 l
|
||||
@ -1746,7 +1735,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/LBA { start_ol
|
||||
/KBA { start_ol
|
||||
1368 5278 m
|
||||
1465 5308 1769 5454 x
|
||||
2074 5600 2383 5680 x
|
||||
@ -1782,7 +1771,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/MBA { start_ol
|
||||
/LBA { start_ol
|
||||
3218 410 m
|
||||
4249 410 l
|
||||
4514 410 4514 210 x
|
||||
@ -1808,7 +1797,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/NBA { start_ol
|
||||
/MBA { start_ol
|
||||
4392 3341 m
|
||||
4392 4176 l
|
||||
5328 4176 l
|
||||
@ -1843,7 +1832,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/OBA { start_ol
|
||||
/NBA { start_ol
|
||||
1850 1802 m
|
||||
1850 0 l
|
||||
914 0 l
|
||||
@ -1878,7 +1867,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/PBA { start_ol
|
||||
/OBA { start_ol
|
||||
1130 4176 m
|
||||
1130 3655 l
|
||||
1377 4017 1613 4168 x
|
||||
@ -1921,7 +1910,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/QBA { start_ol
|
||||
/PBA { start_ol
|
||||
3292 2192 m
|
||||
5134 410 l
|
||||
5224 410 l
|
||||
@ -1962,7 +1951,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/RBA { start_ol
|
||||
/QBA { start_ol
|
||||
2822 -6 m
|
||||
940 3765 l
|
||||
780 3765 l
|
||||
@ -1993,7 +1982,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/SBA { start_ol
|
||||
/RBA { start_ol
|
||||
2922 1166 m
|
||||
3022 1166 l
|
||||
3313 1166 3513 981 x
|
||||
@ -2019,7 +2008,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/TBA { start_ol
|
||||
/SBA { start_ol
|
||||
3020 5105 m
|
||||
3231 5105 3376 4954 x
|
||||
3521 4804 3521 4594 x
|
||||
@ -2046,7 +2035,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/UBA { start_ol
|
||||
/TBA { start_ol
|
||||
5133 6865 m
|
||||
3794 5034 3794 2671 x
|
||||
3794 -2001 l
|
||||
@ -2060,7 +2049,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/VBA { start_ol
|
||||
/UBA { start_ol
|
||||
4874 3506 m
|
||||
4874 2509 l
|
||||
4874 1332 4356 594 x
|
||||
@ -2086,7 +2075,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/WBA { start_ol
|
||||
/VBA { start_ol
|
||||
842 6836 m
|
||||
813 6877 813 6957 x
|
||||
813 7036 877 7096 x
|
||||
@ -2102,7 +2091,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/XBA { start_ol
|
||||
/WBA { start_ol
|
||||
3384 8006 m
|
||||
3794 8006 l
|
||||
3794 -2001 l
|
||||
@ -2111,7 +2100,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/YBA { start_ol
|
||||
/XBA { start_ol
|
||||
2642 -2001 m
|
||||
2232 -2001 l
|
||||
2232 8006 l
|
||||
@ -2120,7 +2109,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/ZBA { start_ol
|
||||
/YBA { start_ol
|
||||
5133 -860 m
|
||||
5162 -901 5162 -981 x
|
||||
5162 -1060 5097 -1120 x
|
||||
@ -2136,7 +2125,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/aBA { start_ol
|
||||
/ZBA { start_ol
|
||||
842 -889 m
|
||||
2181 941 2181 3304 x
|
||||
2181 7977 l
|
||||
@ -2148,9 +2137,65 @@ end_ol
|
||||
813 -1090 813 -1009 x
|
||||
813 -929 842 -889 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
|
||||
} def
|
||||
/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
|
||||
3022 1166 l
|
||||
3313 1166 3513 981 x
|
||||
@ -2165,7 +2210,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/cBA { start_ol
|
||||
/eBA { start_ol
|
||||
5018 5205 m
|
||||
5018 1854 l
|
||||
5018 1009 4439 432 x
|
||||
@ -2196,7 +2241,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/dBA { start_ol
|
||||
/fBA { start_ol
|
||||
5330 873 m
|
||||
5330 785 5160 623 x
|
||||
4990 462 4718 285 x
|
||||
@ -2226,7 +2271,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/eBA { start_ol
|
||||
/gBA { start_ol
|
||||
1634 2736 m
|
||||
1634 410 l
|
||||
3011 410 l
|
||||
@ -2259,7 +2304,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/fBA { start_ol
|
||||
/hBA { start_ol
|
||||
4749 6624 m
|
||||
4073 6624 3646 6294 x
|
||||
3218 5965 3218 5377 x
|
||||
@ -2273,7 +2318,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/gBA { start_ol
|
||||
/iBA { start_ol
|
||||
1226 6573 m
|
||||
957 6573 957 6783 x
|
||||
957 6984 1227 6984 x
|
||||
@ -2287,7 +2332,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/hBA { start_ol
|
||||
/jBA { start_ol
|
||||
3218 8006 m
|
||||
3218 -2001 l
|
||||
2808 -2001 l
|
||||
@ -2296,7 +2341,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/iBA { start_ol
|
||||
/kBA { start_ol
|
||||
4749 -597 m
|
||||
5018 -597 5018 -807 x
|
||||
5018 -1008 4748 -1008 x
|
||||
@ -2310,7 +2355,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/jBA { start_ol
|
||||
/lBA { start_ol
|
||||
1226 -648 m
|
||||
1902 -648 2329 -318 x
|
||||
2757 10 2757 598 x
|
||||
@ -2324,7 +2369,7 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/kBA { start_ol
|
||||
/mBA { start_ol
|
||||
1309 6457 m
|
||||
1955 7428 3181 7428 x
|
||||
4407 7428 5058 6457 x
|
||||
@ -2352,37 +2397,7 @@ end_ol
|
||||
6336 fwd_x
|
||||
end_ol
|
||||
} def
|
||||
/mBA { 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
|
||||
/oBA { start_ol
|
||||
5224 3312 m
|
||||
780 3312 l
|
||||
510 3312 510 3522 x
|
||||
@ -2400,7 +2415,38 @@ end_ol
|
||||
5976 fwd_x
|
||||
end_ol
|
||||
} 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
|
||||
2810 6984 2655 6925 x
|
||||
2499 6867 2436 6804 x
|
||||
@ -2434,6 +2480,40 @@ end_ol
|
||||
3672 6666 3483 6825 x
|
||||
3294 6984 3013 6984 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
|
||||
} def
|
||||
end end
|
||||
@ -2441,50 +2521,60 @@ end end
|
||||
%%EndPrologue
|
||||
%%Page: 1 1
|
||||
paps_bop
|
||||
(+ 36000 807000AAABAACAA+ 64800 807000EAABAAFAA+ 93600 807000GAAHAA+ 115200 807000IAAJAAKAAGAALAAKAAHAAMAA+ 180000 807000GAAJAAIAANAA)paps_exec
|
||||
(>2950OAAPAAQAARAA+ 250700 807000SAATAAUAAVAAWAAQAAKAA+ 308300 807000XAAYAAZAAaAA+ 344300 807000IAAbAA+ 365900 807000cAAIAA)paps_exec
|
||||
(* 515800dAAZAAeAABAA+ 551800 807000IAA)paps_exec
|
||||
(+ 36000 807000AAABAACAA+ 64800 807000EAABAAFAA+ 93600 807000GAAHAA+ 115200 807000IAAHAAJAAIAAKAAJAALAAMAA+ 180000 807000GAAMAAIAANAA)paps_exec
|
||||
(>4750OAAPAAQAARAA+ 254300 807000SAATAAUAAVAAWAAQAAJAA+ 311900 807000XAAYAAZAAaAA+ 347900 807000IAA+ 362300 807000bAAIAA)paps_exec
|
||||
(* 515800cAAZAAdAABAA+ 551800 807000IAA)paps_exec
|
||||
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 768968HBAIBA+ 53928 768968lAAmAAnAA+ 77832 768968iAAoAA+ 90720 768968FBAiAAoAA+ 109584 768968GBA+ 121536 768968HBAIBAFBAiAAoAA+ 152352 768968uAA+ 164304 768968HBAIBA+ 182232 768968lAApAAqAA+ 206136 768968qAAKBAlAAEBADBAlAAxAAqAAnAALBA)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 768968GBAHBA+ 53928 768968kAAlAAmAA+ 77832 768968hAAnAA+ 90720 768968EBAhAAnAA+ 109584 768968FBA+ 121536 768968GBAHBAEBAhAAnAA+ 152352 768968tAA+ 164304 768968GBAHBA+ 182232 768968kAAoAApAA+ 206136 768968pAAJBAkAADBACBAkAAwAApAAmAAKBA)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 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 727712PBAlAAxAApAAtAAQBA+ 77832 727712pAAqAAsAApAAqAArAAqAAmAAxAAlAAxAAtAAwAAmAArAASBA)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 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 727712OBAkAAwAAoAAsAAPBA+ 77832 727712oAApAArAAoAApAAqAApAAlAAwAAkAAwAAsAAvAAlAAqAARBA)paps_exec
|
||||
()paps_exec
|
||||
()paps_exec
|
||||
(+ 36000 696536iAAjAA+ 48816 696536TBA)paps_exec
|
||||
(+ 83808 686456UBA+ 95760 686456vAA+ 107712 686456VBA+ 125640 686456VBA+ 137592 686456WBA)paps_exec
|
||||
(+ 71856 676376HBA+ 83808 676376XBA+ 95760 676376VBA+ 107712 676376VBA+ 125640 676376VBA+ 137592 676376YBA+ 149544 676376lAAmAAnAA)paps_exec
|
||||
(+ 83808 666296ZBA+ 95760 666296VBA+ 107712 666296VBA+ 119664 666296uAAvAA+ 137592 666296aBA)paps_exec
|
||||
(+ 36000 655280iAAoAA+ 48888 655280TBA)paps_exec
|
||||
(+ 83808 645200UBA+ 95760 645200VBA+ 107712 645200vAA+ 119664 645200VBA+ 131616 645200WBA)paps_exec
|
||||
(+ 71856 635120HBA+ 83808 635120XBA+ 95760 635120vAA+ 107712 635120VBA+ 119664 635120vAA+ 131616 635120YBAbBA)paps_exec
|
||||
(+ 83808 625040ZBA+ 95760 625040VBA+ 107712 625040vAA+ 119664 625040VBA+ 131616 625040aBA)paps_exec
|
||||
(+ 36000 696536hAAiAA+ 48816 696536SBA)paps_exec
|
||||
(+ 71856 686456GBA+ 83808 686456TBA+ 95760 686456uAA+ 107712 686456UBA+ 125640 686456UBA+ 137592 686456VBA)paps_exec
|
||||
(+ 83808 676376WBA+ 95760 676376UBA+ 107712 676376UBA+ 125640 676376UBA+ 137592 676376XBA+ 149544 676376kAAlAAmAA)paps_exec
|
||||
(+ 83808 666296YBA+ 95760 666296UBA+ 107712 666296UBA+ 119664 666296tAAuAA+ 137592 666296ZBA)paps_exec
|
||||
(+ 36000 655280hAAnAA+ 48888 655280SBA)paps_exec
|
||||
(+ 71856 645200aBAGBAaBA+ 89784 645200TBA+ 101736 645200UBA+ 113688 645200uAA+ 125640 645200UBA+ 137592 645200VBA)paps_exec
|
||||
(+ 71856 635120bBAcBA+ 89784 635120WBA+ 101736 635120uAA+ 113688 635120UBA+ 125640 635120uAA+ 137592 635120XBAdBA)paps_exec
|
||||
(+ 89784 625040YBA+ 101736 625040UBA+ 113688 625040uAA+ 125640 625040UBA+ 137592 625040ZBA)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 593864wAAsAAqAApAAlAAxAAwAApAABBA+ 95760 593864xAAyAAqAA+ 119664 593864qAAQBAsAApAAqAArAArAAtAAwAAmAA+ 185400 593864iAAjAA+ 198216 593864FBAiAAjAA+ 217008 593864GBA+ 228960 593864HBAIBAFBAiAAjAA+ 259704 593864uAA+ 271656 593864HBAIBA+ 289584 593864TBA)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 593864vAArAApAAoAAkAAwAAvAAoAAABA+ 95760 593864wAAxAApAA+ 119664 593864pAAPBArAAoAApAAqAAqAAsAAvAAlAA+ 185400 593864hAAiAA+ 198216 593864EBAhAAiAA+ 217008 593864FBA+ 228960 593864GBAHBAEBAhAAiAA+ 259704 593864tAA+ 271656 593864GBAHBA+ 289584 593864SBA)paps_exec
|
||||
()paps_exec
|
||||
(+ 71856 573704UBA+ 83808 573704vAA+ 95760 573704VBA+ 113688 573704VBA+ 125640 573704WBA+ 137592 573704fBA+ 161496 573704UBA+ 173448 573704vAA+ 185400 573704VBA+ 203328 573704VBA+ 215280 573704WBA+ 263088 573704UBA+ 275040 573704vAA+ 286992 573704VBA+ 298944 573704VBA+ 310896 573704WBA+ 322848 573704gBA+ 334800 573704fBA+ 358704 573704UBA+ 370656 573704vAA+ 382608 573704VBA+ 400536 573704VBA+ 412488 573704WBA+ 454320 573704UBA+ 466272 573704vAA+ 478224 573704VBA+ 490176 573704VBA+ 502128 573704WBA+ 514080 573704gBA)paps_exec
|
||||
(+ 59904 563624HBA+ 71856 563624XBA+ 83808 563624VBA+ 95760 563624VBA+ 113688 563624VBA+ 125640 563624YBA+ 137592 563624hBA+ 149544 563624HBA+ 161496 563624XBA+ 173448 563624VBA+ 185400 563624VBA+ 203328 563624VBA+ 215280 563624YBA+ 233208 563624GBA+ 251136 563624HBA+ 263088 563624XBA+ 275040 563624VBA+ 286992 563624vAA+ 298944 563624VBA+ 310896 563624YBA+ 322848 563624hBA+ 334800 563624hBA+ 346752 563624HBA+ 358704 563624XBA+ 370656 563624VBA+ 382608 563624VBA+ 400536 563624VBA+ 412488 563624YBA+ 430416 563624uAA+ 442368 563624HBA+ 454320 563624XBA+ 466272 563624VBA+ 478224 563624vAA+ 490176 563624VBA+ 502128 563624YBA+ 514080 563624hBABBA)paps_exec
|
||||
(+ 71856 553544ZBA+ 83808 553544VBA+ 95760 553544VBA+ 107712 553544uAAvAA+ 125640 553544aBA+ 137592 553544iBA+ 161496 553544ZBA+ 173448 553544VBA+ 185400 553544VBA+ 197352 553544uAAvAA+ 215280 553544aBA+ 263088 553544ZBA+ 275040 553544VBA+ 286992 553544VBA+ 298944 553544vAA+ 310896 553544aBA+ 322848 553544jBA+ 334800 553544iBA+ 358704 553544ZBA+ 370656 553544VBA+ 382608 553544VBA+ 394560 553544uAAvAA+ 412488 553544aBA+ 454320 553544ZBA+ 466272 553544VBA+ 478224 553544VBA+ 490176 553544vAA+ 502128 553544aBA+ 514080 553544jBA)paps_exec
|
||||
(+ 59904 573704GBA+ 71856 573704TBA+ 83808 573704uAA+ 95760 573704UBA+ 113688 573704UBA+ 125640 573704VBA+ 137592 573704hBA+ 149544 573704GBA+ 161496 573704TBA+ 173448 573704uAA+ 185400 573704UBA+ 203328 573704UBA+ 215280 573704VBA+ 251136 573704GBA+ 263088 573704TBA+ 275040 573704uAA+ 286992 573704UBA+ 298944 573704UBA+ 310896 573704VBA+ 322848 573704iBA+ 334800 573704hBA+ 346752 573704GBA+ 358704 573704TBA+ 370656 573704uAA+ 382608 573704UBA+ 400536 573704UBA+ 412488 573704VBA+ 442368 573704GBA+ 454320 573704TBA+ 466272 573704uAA+ 478224 573704UBA+ 490176 573704UBA+ 502128 573704VBA+ 514080 573704iBA)paps_exec
|
||||
(+ 71856 563624WBA+ 83808 563624UBA+ 95760 563624UBA+ 113688 563624UBA+ 125640 563624XBA+ 137592 563624jBA+ 161496 563624WBA+ 173448 563624UBA+ 185400 563624UBA+ 203328 563624UBA+ 215280 563624XBA+ 233208 563624FBA+ 263088 563624WBA+ 275040 563624UBA+ 286992 563624uAA+ 298944 563624UBA+ 310896 563624XBA+ 322848 563624jBA+ 334800 563624jBA+ 358704 563624WBA+ 370656 563624UBA+ 382608 563624UBA+ 400536 563624UBA+ 412488 563624XBA+ 430416 563624tAA+ 454320 563624WBA+ 466272 563624UBA+ 478224 563624uAA+ 490176 563624UBA+ 502128 563624XBA+ 514080 563624jBAABA)paps_exec
|
||||
(+ 71856 553544YBA+ 83808 553544UBA+ 95760 553544UBA+ 107712 553544tAAuAA+ 125640 553544ZBA+ 137592 553544kBA+ 161496 553544YBA+ 173448 553544UBA+ 185400 553544UBA+ 197352 553544tAAuAA+ 215280 553544ZBA+ 263088 553544YBA+ 275040 553544UBA+ 286992 553544UBA+ 298944 553544uAA+ 310896 553544ZBA+ 322848 553544lBA+ 334800 553544kBA+ 358704 553544YBA+ 370656 553544UBA+ 382608 553544UBA+ 394560 553544tAAuAA+ 412488 553544ZBA+ 454320 553544YBA+ 466272 553544UBA+ 478224 553544UBA+ 490176 553544uAA+ 502128 553544ZBA+ 514080 553544lBA)paps_exec
|
||||
()paps_exec
|
||||
(+ 36000 531800CBAyAAtAAdBAyAA+ 71856 531800rAAtAAPBAsAAEBAtAAgAAtAAqAArAA+ 137592 531800xAAwAA+ 155520 531800xAAyAAqAA+ 179424 531800PBAlAAxAApAAtAAQBA+ 221256 531800PBADBAEBAxAAtAAsAAEBAtAAdBAlAAxAAtAAwAAmAA+ 310896 531800wAAsAAqAApAAlAAxAAtAAwAAmAABBA+ 376632 531800CBAyAAqAApAAqAA+ 412488 531800kBA+ 421992 531800pAAqAAsAApAAqAArAAqAAmAAxAArAA+ 487728 531800xAAyAAqAA+ 511632 531800VBA)paps_exec
|
||||
(+ 36000 521720PBAlAAxAApAAtAAQBABBA)paps_exec
|
||||
(+ 36000 531800BBAxAAsAAfBAxAA+ 71856 531800qAAsAAOBArAADBAsAAfAAsAApAAqAA+ 137592 531800wAAvAA+ 155520 531800wAAxAApAA+ 179424 531800OBAkAAwAAoAAsAAPBA+ 221256 531800OBACBADBAwAAsAArAADBAsAAfBAkAAwAAsAAvAAlAA+ 310896 531800vAArAApAAoAAkAAwAAsAAvAAlAAABA+ 376632 531800BBAxAApAAoAApAA+ 412488 531800mBA+ 421992 531800oAApAArAAoAApAAqAApAAlAAwAAqAA+ 487728 531800wAAxAApAA+ 511632 531800UBA)paps_exec
|
||||
(+ 36000 521720OBAkAAwAAoAAsAAPBAABA)paps_exec
|
||||
()paps_exec
|
||||
(+ 71856 501560UBA+ 83808 501560vAA+ 95760 501560VBA+ 113688 501560VBA+ 125640 501560WBA+ 149544 501560UBA+ 161496 501560mBA+ 173448 501560VBA+ 185400 501560VBA+ 197352 501560WBA+ 221256 501560UBA+ 233208 501560VBA+ 251136 501560VBA+ 263088 501560VBA+ 275040 501560WBA)paps_exec
|
||||
(+ 59904 489896HBA+ 71856 489896XBA+ 83808 489896VBA+ 95760 489896VBA+ 113688 489896VBA+ 125640 489896YBA+ 137592 489896HBA+ 149544 489896XBA+ 161496 489896VBA+ 173448 489896vAA+ 185400 489896VBA+ 197352 489896YBA+ 209304 489896HBA+ 221256 489896XBA+ 233208 489896VBA+ 245160 489896uAAvAA+ 263088 489896VBA+ 275040 489896YBA+ 286992 489896nBA+ 298944 489896kBAbBA)paps_exec
|
||||
(+ 71856 479816ZBA+ 83808 479816VBA+ 95760 479816VBA+ 107712 479816uAAvAA+ 125640 479816aBA+ 149544 479816ZBA+ 161496 479816VBA+ 173448 479816VBA+ 185400 479816VBA+ 197352 479816aBA+ 221256 479816ZBA+ 233208 479816VBA+ 251136 479816VBA+ 263088 479816VBA+ 275040 479816aBA)paps_exec
|
||||
(+ 59904 501560GBA+ 71856 501560TBA+ 83808 501560uAA+ 95760 501560UBA+ 113688 501560UBA+ 125640 501560VBA+ 137592 501560GBA+ 149544 501560TBA+ 161496 501560cBA+ 173448 501560UBA+ 185400 501560UBA+ 197352 501560VBA+ 209304 501560GBA+ 221256 501560TBA+ 233208 501560UBA+ 251136 501560UBA+ 263088 501560UBA+ 275040 501560VBA)paps_exec
|
||||
(+ 71856 489896WBA+ 83808 489896UBA+ 95760 489896UBA+ 113688 489896UBA+ 125640 489896XBA+ 149544 489896WBA+ 161496 489896UBA+ 173448 489896uAA+ 185400 489896UBA+ 197352 489896XBA+ 221256 489896WBA+ 233208 489896UBA+ 245160 489896tAAuAA+ 263088 489896UBA+ 275040 489896XBA+ 286992 489896oBA+ 298944 489896mBAdBA)paps_exec
|
||||
(+ 71856 479816YBA+ 83808 479816UBA+ 95760 479816UBA+ 107712 479816tAAuAA+ 125640 479816ZBA+ 149544 479816YBA+ 161496 479816UBA+ 173448 479816UBA+ 185400 479816UBA+ 197352 479816ZBA+ 221256 479816YBA+ 233208 479816UBA+ 251136 479816UBA+ 263088 479816UBA+ 275040 479816ZBA)paps_exec
|
||||
()paps_exec
|
||||
(+ 36000 458072MBAyAAqAA+ 59904 458072PBADBAEBAxAAtAAsAAEBAtAAdBAlAAxAAtAAwAAmAA+ 149544 458072wAAsAAqAApAAlAAxAAtAAwAAmAA+ 209304 458072lAAsAAsAAlAApAAqAAmAAxAAEBARBA+ 275040 458072pAAqAAxAADBApAAmAArAA+ 322848 458072kBA+ 332352 458072ABAqAAdBAlAADBArAAqAA+ 380160 458072xAAyAAqAA+ 404064 458072xAAyAAtAApAAnAA+ 439920 458072gAAlAAdBAxAAwAApAA+ 481752 458072CBAtAAEBAEBA)paps_exec
|
||||
(+ 36000 447992mAADBAEBAEBAtAAgAARBA+ 83808 447992lAAmAARBA+ 107712 447992xAAqAApAAPBArAA+ 143568 447992ABAqAArAAtAAnAAqAArAA+ 191376 447992dBAqAAmAAxAAqAApAA+ 233208 447992xAAqAApAAPBArAABBA+ 275040 447992lAAmAAnAA+ 298944 447992xAAyAAqAA+ 322848 447992gAAtAApAArAAxAA+ 358704 447992gAAlAAdBAxAAwAApAA+ 400536 447992CBAtAAEBAEBA+ 430416 447992mAADBAEBAEBAtAAgAARBA+ 478224 447992lAAmAARBA+ 502128 447992dBAqAAmAAxAAqAApAA)paps_exec
|
||||
(+ 36000 437912xAAqAApAAPBArAAbBA)paps_exec
|
||||
(+ 36000 458072LBAxAApAA+ 59904 458072OBACBADBAwAAsAArAADBAsAAfBAkAAwAAsAAvAAlAA+ 149544 458072vAArAApAAoAAkAAwAAsAAvAAlAA+ 209304 458072kAArAArAAkAAoAApAAlAAwAADBAQBA+ 275040 458072oAApAAwAACBAoAAlAAqAA+ 322848 458072mBA+ 332352 458072zAApAAfBAkAACBAqAApAA+ 380160 458072wAAxAApAA+ 404064 458072wAAxAAsAAoAAmAA+ 439920 458072fAAkAAfBAwAAvAAoAA+ 481752 458072BBAsAADBADBA)paps_exec
|
||||
(+ 36000 447992lAACBADBADBAsAAfAAQBA+ 83808 447992kAAlAAQBA+ 107712 447992wAApAAoAAOBAqAA+ 143568 447992zAApAAqAAsAAmAApAAqAA+ 191376 447992fBApAAlAAwAApAAoAA+ 233208 447992wAApAAoAAOBAqAAABA+ 275040 447992kAAlAAmAA+ 298944 447992wAAxAApAA+ 322848 447992fAAsAAoAAqAAwAA+ 358704 447992fAAkAAfBAwAAvAAoAA+ 400536 447992BBAsAADBADBA+ 430416 447992lAACBADBADBAsAAfAAQBA+ 478224 447992kAAlAAQBA+ 502128 447992fBApAAlAAwAApAAoAA)paps_exec
|
||||
(+ 36000 437912wAApAAoAAOBAqAAdBA)paps_exec
|
||||
()paps_exec
|
||||
(+ 36000 416816iAAtAAPBAtAAEBAlAApAAEBARBABBA+ 101736 416816iAAoAA+ 114624 416816FBAiAAoAA+ 133488 416816GBA+ 145440 416816HBAIBAFBAiAAoAA+ 176256 416816uAA+ 188208 416816HBAIBA+ 206136 416816TBA)paps_exec
|
||||
(+ 36000 416816hAAsAAOBAsAADBAkAAoAADBAQBAABA+ 101736 416816hAAnAA+ 114624 416816EBAhAAnAA+ 133488 416816FBA+ 145440 416816GBAHBAEBAhAAnAA+ 176256 416816tAA+ 188208 416816GBAHBA+ 206136 416816SBA)paps_exec
|
||||
()paps_exec
|
||||
(+ 77832 396656UBA+ 89784 396656VBA+ 101736 396656vAA+ 113688 396656VBA+ 125640 396656WBA+ 137592 396656UBA+ 149544 396656vAA+ 161496 396656vAA+ 173448 396656VBA+ 185400 396656WBA+ 197352 396656UBA+ 209304 396656uAAvAA+ 233208 396656vAA+ 251136 396656VBA+ 263088 396656WBA+ 304920 396656UBA+ 316872 396656vAA+ 328824 396656vAA+ 340776 396656vAA+ 352728 396656WBA+ 364680 396656UBA+ 376632 396656uAAvAA+ 400536 396656vAA+ 418464 396656VBA+ 430416 396656WBA+ 472248 396656UBA+ 484200 396656VBA+ 496152 396656vAA+ 508104 396656VBA+ 520056 396656WBA)paps_exec
|
||||
(+ 59904 386576HBAoBA+ 77832 386576XBA+ 89784 386576vAA+ 101736 386576VBA+ 113688 386576vAA+ 125640 386576YBA+ 137592 386576XBA+ 149544 386576vAA+ 161496 386576vAA+ 173448 386576vAA+ 185400 386576YBA+ 197352 386576XBA+ 215280 386576vAA+ 227232 386576uAAvAA+ 251136 386576vAA+ 263088 386576YBA+ 275040 386576nBA+ 286992 386576HBAoBA+ 304920 386576XBA+ 316872 386576vAA+ 328824 386576mBA+ 340776 386576vAA+ 352728 386576YBA+ 364680 386576XBA+ 382608 386576vAA+ 394560 386576uAAvAA+ 418464 386576vAA+ 430416 386576YBA+ 442368 386576nBA+ 454320 386576HBAoBA+ 472248 386576XBA+ 484200 386576vAA+ 496152 386576VBA+ 508104 386576vAA+ 520056 386576YBAbBA)paps_exec
|
||||
(+ 77832 376496ZBA+ 89784 376496VBA+ 101736 376496vAA+ 113688 376496VBA+ 125640 376496aBA+ 137592 376496ZBA+ 149544 376496VBA+ 161496 376496vAA+ 173448 376496vAA+ 185400 376496aBA+ 197352 376496ZBA+ 215280 376496VBA+ 233208 376496vAA+ 245160 376496uAAvAA+ 263088 376496aBA+ 304920 376496ZBA+ 316872 376496vAA+ 328824 376496vAA+ 340776 376496vAA+ 352728 376496aBA+ 364680 376496ZBA+ 382608 376496VBA+ 400536 376496vAA+ 412488 376496uAAvAA+ 430416 376496aBA+ 472248 376496ZBA+ 484200 376496VBA+ 496152 376496vAA+ 508104 376496VBA+ 520056 376496aBA)paps_exec
|
||||
(+ 53928 396656aBAGBAaBA+ 77832 396656TBA+ 89784 396656UBA+ 101736 396656uAA+ 113688 396656UBA+ 125640 396656VBA+ 137592 396656aBAGBAaBA+ 155520 396656TBA+ 167472 396656bBAcBA+ 191376 396656uAA+ 209304 396656UBA+ 227232 396656VBA+ 239184 396656aBAGBAaBA+ 257112 396656TBA+ 269064 396656tAAbBAcBA+ 304920 396656uAA+ 328824 396656UBA+ 340776 396656VBA)paps_exec
|
||||
(+ 53928 386576bBAcBA+ 77832 386576WBA+ 89784 386576uAA+ 101736 386576UBA+ 113688 386576uAA+ 125640 386576XBA+ 137592 386576bBAcBA+ 155520 386576WBA+ 173448 386576uAA+ 185400 386576bBAcBA+ 209304 386576uAA+ 227232 386576XBA+ 239184 386576bBAcBA+ 257112 386576WBA+ 281016 386576uAA+ 292968 386576tAAbBAcBA+ 328824 386576uAA+ 340776 386576XBAdBA)paps_exec
|
||||
(+ 77832 376496YBA+ 89784 376496UBA+ 101736 376496uAA+ 113688 376496UBA+ 125640 376496ZBA+ 155520 376496YBA+ 173448 376496UBA+ 191376 376496uAA+ 203328 376496bBAcBA+ 227232 376496ZBA+ 257112 376496YBA+ 281016 376496UBA+ 304920 376496uAA+ 316872 376496tAAbBAcBA+ 340776 376496ZBA)paps_exec
|
||||
()paps_exec
|
||||
(+ 36000 356336pBApAAoAAfAAvAAoAAOBAsAAlAAMBA+ 101736 356336wAAxAApAA+ 125640 356336OBACBADBAwAAsAArAADBAsAAfBAkAAwAAsAAvAAlAA+ 215280 356336vAArAApAAoAAkAAwAAsAAvAAlAA+ 275040 356336vAAlAA+ 292968 356336wAAxAApAA+ 316872 356336DBAkAAqAAwAA+ 346752 356336wAABBAvAA+ 370656 356336OBAkAAwAAoAAsAAfBApAAqAA+ 424440 356336oAApAAwAACBAoAAlAAqAA+ 472248 356336wAAxAApAA+ 496152 356336pAAPBArAAoAApAAqAAqAAsAAvAAlAA)paps_exec
|
||||
()paps_exec
|
||||
(+ 47952 336176aBAGBAaBAqBAaBA+ 77832 336176TBA+ 89784 336176UBA+ 101736 336176uAA+ 113688 336176UBA+ 125640 336176VBA+ 137592 336176TBA+ 149544 336176tAAuAA+ 173448 336176UBA+ 191376 336176uAA+ 203328 336176VBA)paps_exec
|
||||
(+ 53928 324512cBAbBAcBA+ 77832 324512WBA+ 89784 324512uAA+ 101736 324512UBA+ 113688 324512uAA+ 125640 324512XBA+ 137592 324512WBA+ 155520 324512UBA+ 173448 324512UBA+ 191376 324512UBA+ 203328 324512XBA+ 215280 324512oBA+ 227232 324512mBAdBA)paps_exec
|
||||
(+ 77832 314432YBA+ 89784 314432UBA+ 101736 314432uAA+ 113688 314432UBA+ 125640 314432ZBA+ 137592 314432YBA+ 155520 314432uAA+ 173448 314432UBA+ 185400 314432tAAuAA+ 203328 314432ZBA)paps_exec
|
||||
()paps_exec
|
||||
(+ 36000 292688eAAwAA+ 53928 292688sAAqAA+ 71856 292688rBACBAsAAwAApAA+ 107712 292688vAAzAAJBAsAAvAACBAqAA+ 155520 292688wAAxAAkAAwAA+ 185400 292688wAAxAAsAAqAA+ 215280 292688vAArAApAAoAAkAAwAAsAAvAAlAA+ 275040 292688oAApAAwAACBAoAAlAAqAA+ 322848 292688mBA+ 332352 292688qAAsAAlAAfBApAA+ 368208 292688wAAxAApAAoAApAA+ 404064 292688kAAoAApAA+ 427968 292688lAAvAA+ 445896 292688fBAvAAOBArAAvAAlAApAAlAAwAAqAA+ 511632 292688wAAxAAkAAwAA)paps_exec
|
||||
(+ 36000 282608BBAsAADBADBA+ 65880 282608lAAvAAwAA+ 89784 282608OBAkAAwAAfBAxAA+ 125640 282608BBAsAAwAAxAA+ 155520 282608kAA+ 167472 282608UBA+ 179424 282608wAAxAAoAAvAACBAMBAxAAvAACBAwAA+ 245160 282608wAAxAApAA+ 269064 282608OBACBADBAwAAsAArAADBAsAAfBAkAAwAAsAAvAAlAA+ 358704 282608vAAfAA+ 376632 282608wAAxAApAAqAApAA+ 412488 282608OBAkAAwAAoAAsAAfBApAAqAAdBA+ 472248 282608LBAxAApAAoAApAAfAAvAAoAApAAABA)paps_exec
|
||||
(+ 36000 270944wAAxAApAA+ 59904 270944qAApAAfBAvAAlAAmAA+ 101736 270944pAAPBArAAoAApAAqAAqAAsAAvAAlAA+ 167472 270944sAAqAA+ 185400 270944kAADBAqAAvAA+ 215280 270944pAArBACBAsAAJBAkAADBApAAlAAwAA+ 281016 270944wAAvAA+ 298944 270944wAAxAApAA+ 322848 270944yAApAAoAAvAA+ 352728 270944OBAkAAwAAoAAsAAPBA+ 394560 270944mBAdBA)paps_exec
|
||||
paps_eop
|
||||
showpage
|
||||
%%Pages: 1
|
||||
|
||||
@ -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
|
||||
|
||||
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
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
2786
solutions/exam1/prob3.ps
Normal file
2786
solutions/exam1/prob3.ps
Normal file
File diff suppressed because it is too large
Load Diff
47
solutions/exam1/prob4
Normal file
47
solutions/exam1/prob4
Normal file
@ -0,0 +1,47 @@
|
||||
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
|
||||
|
||||
|
||||
2631
solutions/exam1/prob4.ps
Normal file
2631
solutions/exam1/prob4.ps
Normal file
File diff suppressed because it is too large
Load Diff
Loading…
Reference in New Issue
Block a user