diff --git a/solutions/exam1/.ps b/solutions/exam1/.ps new file mode 100644 index 0000000..e69de29 diff --git a/solutions/exam1/prob1 b/solutions/exam1/prob1 index 29dbf9e..442e2bc 100644 --- a/solutions/exam1/prob1 +++ b/solutions/exam1/prob1 @@ -1,37 +1,33 @@ -The characteristic equation for the spin operator S𝓏 is +If S𝓏 and S𝓍 are spin-1 operators in the z basis, what are the results if S𝓏 (S𝓏 + Δ§)(S𝓏 - Δ§) and S𝓍 (S𝓍 + Δ§)(S𝓍 - Δ§) are evaluated? - S𝓏 (S𝓏 + Δ§)(S𝓏 - Δ§) = 0. - -The eigenvalues, which are the roots of this equation, are - - Ξ» = 0, Β±Δ§. - -The S𝓏 operator is already diagonalized in its own basis, so the matrix has the immediately constructable form of - - S𝓏 ≐ - βŽ› 1 0 0 ⎞ - Δ§ ⎜ 0 0 0 ⎟ - ⎝ 0 0 -1 ⎠ +The spin eigenstates and eigenvalues are known from experiment for a spin-1 system, and the spin-z and spin-x operators in the z basis, S𝓏 and S𝓍, have the following matrix representations: -To produce the operator S𝓏, one can apply the rotation matrix for a rotation about the y axis., where the angle of rotation is Ο€/2. +S𝓏 ≐ + βŽ› 1 0 0 ⎞ + Δ§ ⎜ 0 0 0 ⎟ and + ⎝ 0 0 -1 ⎠ +S𝓍 ≐ + βŽ› 0 1 0 ⎞ + Δ§ ⎜ 1 0 1 ⎟. + ⎝ 0 1 0 ⎠ - βŽ› cosΞΈ 0 sinΞΈ ⎞ βŽ› 0 0 1 ⎞ - R = ⎜ 0 1 0 ⎟ = ⎜ 0 1 0 ⎟ - ⎝-sinΞΈ 0 cosΞΈ ⎠ ⎝-1 0 0 ⎠ +Using the matrix representations, the expressions can be evaluated. For the spin-z operator, the expression S𝓏 (S𝓏 + Δ§)(S𝓏 - Δ§) ≐ -S𝓍 = S𝓏 R ≐ - βŽ› 1 0 0 ⎞ βŽ› 0 0 1 ⎞ βŽ› 0 0 1 ⎞ - Δ§ ⎜ 0 0 0 ⎟ ⎜ 0 1 0 ⎟ = ⎜ 0 1 0 ⎟ - ⎝ 0 0 -1 ⎠ ⎝-1 0 0 ⎠ ⎝-1 0 0 ⎠ - - - - - -The same is true for the S𝓍 operator in its basis. To express this operator in the z basis, however, it must be diagonalized. + βŽ› 1 0 0 ⎞ ⎧ βŽ› 1 0 0 ⎞ βŽ› 1 0 0 ⎞ ⎫ ⎧ βŽ› 1 0 0 ⎞ βŽ› 1 0 0 ⎞ ⎫ + Δ§ ⎜ 0 0 0 ⎟ βŽͺ Δ§ ⎜ 0 0 0 ⎟ + Δ§ ⎜ 0 1 0 ⎟ βŽͺ βŽͺ Δ§ ⎜ 0 0 0 ⎟ - Δ§ ⎜ 0 1 0 ⎟ βŽͺ, + ⎝ 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, + βŽ› 1 0 0 ⎞ βŽ› 2 0 0 ⎞ βŽ› 0 0 0 ⎞ + Δ§ ⎜ 0 0 0 ⎟ Δ§ ⎜ 0 1 0 ⎟ Δ§ ⎜ 0 -1 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. +Similarly, S𝓍 (S𝓍 + Δ§)(S𝓍 - Δ§) ≐ + βŽ› 0 1 0 ⎞ βŽ› 1 1 0 ⎞ βŽ› -1 1 0 ⎞ βŽ› 1 1 1 ⎞ βŽ› -1 1 0 ⎞ βŽ› 0 1 0 ⎞ + Δ§Β³ ⎜ 1 0 1 ⎟ ⎜ 1 1 1 ⎟ ⎜ 1 -1 1 ⎟ = Δ§Β³ ⎜ 1 2 1 ⎟ ⎜ 1 -1 1 ⎟ = Δ§Β³ ⎜ 1 0 1 ⎟. + ⎝ 0 1 0 ⎠ ⎝ 0 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 1 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 0 1 0 ⎠ \ No newline at end of file diff --git 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end_ol } def /NBA { start_ol -1716 7260 m -1716 3974 l -2605 5220 3907 5220 x -5011 5220 5775 4444 x -6540 3669 6540 2543 x -6540 1410 5770 615 x -4999 -180 3907 -180 x -2568 -180 1716 1063 x -1716 0 l -547 0 l -216 0 216 263 x -216 492 552 492 x -1224 492 l -1224 6768 l -552 6768 l -216 6768 216 7020 x -216 7260 547 7260 x -1716 7260 l -3882 4727 m -2982 4727 2349 4083 x -1716 3441 1716 2518 x -1716 1596 2349 954 x -2982 312 3882 312 x -4770 312 5409 951 x -6048 1590 6048 2482 x -6048 3429 5421 4078 x -4794 4727 3882 4727 x -7200 fwd_x +2559 327 m +3361 327 l +3567 327 3567 167 x +3567 0 3360 0 x +1431 0 l +1224 0 1224 167 x +1224 327 1429 327 x +2232 327 l +2232 4208 l +903 4208 l +903 3624 l +903 3408 735 3408 x +576 3408 576 3624 x +576 4536 l +4215 4536 l +4215 3624 l +4215 3408 4055 3408 x +3983 3408 3935 3468 x +3888 3529 3888 3624 x +3888 4208 l +2559 4208 l +2559 327 l +4824 fwd_x end_ol } def /OBA { start_ol -1356 5040 m -1356 4414 l -1654 4854 1940 5036 x -2227 5220 2608 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2984 l +72 2984 72 3150 x +72 3312 294 3312 x +903 3312 l +4824 fwd_x end_ol } def /RBA { start_ol -3625 6170 m -3877 6170 4052 5989 x -4226 5809 4226 5557 x -4226 5316 4045 5142 x -3865 4968 3624 4968 x -3385 4968 3204 5148 x -3024 5329 3024 5568 x -3024 5809 3204 5989 x -3385 6170 3625 6170 x -6276 3240 m -938 3240 l -613 3240 613 3492 x -613 3732 938 3732 x -6276 3732 l -6601 3732 6601 3492 x -6601 3240 6276 3240 x -6276 1512 m -938 1512 l -613 1512 613 1764 x -613 2004 938 2004 x -6276 2004 l -6601 2004 6601 1764 x -6601 1512 6276 1512 x -7200 fwd_x +2629 1739 m +4099 327 l +4171 327 l +4387 327 4387 167 x +4387 0 4171 0 x +3117 0 l +2901 0 2901 167 x +2901 327 3117 327 x +3636 327 l +2397 1509 l +1143 327 l +1686 327 l +1902 327 1902 167 x +1902 0 1686 0 x +623 0 l +407 0 407 167 x +407 327 623 327 x +695 327 l +2165 1739 l +862 2984 l +807 2984 l +591 2984 591 3152 x +591 3312 807 3312 x +1694 3312 l +1910 3312 1910 3152 x +1910 2984 1694 2984 x +1326 2984 l +2397 1953 l +3492 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+4824 fwd_x +end_ol + } def +/hBA { start_ol +1039 5288 m +824 5288 824 5456 x +824 5616 1040 5616 x +1712 5616 2152 5263 x +2592 4912 2592 4289 x +2592 -1576 l +2264 -1576 l +2264 4289 l +2264 4761 1922 5024 x +1580 5288 1039 5288 x +4824 fwd_x +end_ol + } def +/iBA { start_ol +2559 6393 m +2559 -1598 l +2232 -1598 l +2232 6393 l +2559 6393 l +4824 fwd_x +end_ol + } def +/jBA { start_ol +3784 -464 m +3999 -464 3999 -632 x +3999 -792 3783 -792 x +3111 -792 2671 -439 x +2232 -88 2232 534 x +2232 6400 l +2559 6400 l +2559 534 l +2559 63 2901 -200 x +3243 -464 3784 -464 x +4824 fwd_x +end_ol + } def +/kBA { start_ol +1039 -432 m +1580 -432 1922 -168 x +2264 93 2264 563 x +2264 6408 l +2592 6408 l +2592 562 l +2592 -58 2152 -408 x +1712 -759 1040 -759 x +824 -759 824 -599 x +824 -432 1039 -432 x +4824 fwd_x +end_ol + } def +/lBA { start_ol +1046 5157 m +1561 5932 2540 5932 x +3520 5932 4038 5157 x +4558 4381 4558 2909 x +4558 1436 4038 661 x +3520 -113 2540 -113 x +1561 -113 1046 661 x 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2664 x +4171 1512 m +623 1512 l +407 1512 407 1679 x +407 1839 623 1839 x +4171 1839 l +4387 1839 4387 1679 x +4387 1512 4171 1512 x +4824 fwd_x +end_ol + } def +/pBA { start_ol +2436 5544 m +2277 5544 2155 5496 x +2035 5450 1985 5400 x +1936 5349 1886 5302 x +1838 5256 1792 5256 x +1739 5256 1697 5295 x +1656 5334 1656 5389 x +1656 5539 1919 5685 x +2183 5832 2444 5832 x +2781 5832 3010 5620 x +3240 5410 3240 5099 x +3240 4758 2872 4512 x +3312 4224 3312 3808 x +3312 3456 3027 3204 x +2743 2952 2344 2952 x +2048 2952 1740 3096 x +1433 3240 1433 3376 x +1433 3432 1472 3471 x +1513 3512 1569 3512 x +1593 3512 1824 3376 x +2056 3240 2344 3240 x +2615 3240 2819 3407 x +3024 3576 3024 3808 x +3024 4047 2824 4215 x +2626 4384 2320 4392 x +2160 4407 2160 4536 x +2160 4623 2223 4651 x +2286 4680 2450 4680 x +2669 4680 2810 4798 x +2952 4917 2952 5099 x +2952 5290 2803 5416 x +2656 5544 2436 5544 x +4824 fwd_x end_ol } def end end @@ -2222,50 +2467,47 @@ end end %%EndPrologue %%Page: 1 1 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