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41 lines
2.1 KiB
Plaintext
41 lines
2.1 KiB
Plaintext
If S𝓏 and S𝓍 are spin-1 operators in the z basis, what are the results if S𝓏 (S𝓏 + ħ)(S𝓏 - ħ) and S𝓍 (S𝓍 + ħ)(S𝓍 - ħ) are evaluated?
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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:
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S𝓏 ≐
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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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͟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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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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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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͟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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Performing the multiplication operation on the last two matrices returns the expression
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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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Because each pair of row and column in this matrix has alternating 0s and ±1s, every multiplication operation will return 0. The second expression is therefore equivalent to the zero matrix 𝟘. |