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38 lines
1.0 KiB
Plaintext
38 lines
1.0 KiB
Plaintext
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The characteristic equation for the spin operator S𝓏 is
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S𝓏 (S𝓏 + ħ)(S𝓏 - ħ) = 0.
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The eigenvalues, which are the roots of this equation, are
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λ = 0, ±ħ.
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The S𝓏 operator is already diagonalized in its own basis, so the matrix has the immediately constructable form of
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S𝓏 ≐
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⎛ 1 0 0 ⎞
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ħ ⎜ 0 0 0 ⎟
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⎝ 0 0 -1 ⎠
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To produce the operator S𝓏, one can apply the rotation matrix for a rotation about the y axis., where the angle of rotation is π/2.
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⎛ cosθ 0 sinθ ⎞ ⎛ 0 0 1 ⎞
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R = ⎜ 0 1 0 ⎟ = ⎜ 0 1 0 ⎟
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⎝-sinθ 0 cosθ ⎠ ⎝-1 0 0 ⎠
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S𝓍 = S𝓏 R ≐
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⎛ 1 0 0 ⎞ ⎛ 0 0 1 ⎞ ⎛ 0 0 1 ⎞
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ħ ⎜ 0 0 0 ⎟ ⎜ 0 1 0 ⎟ = ⎜ 0 1 0 ⎟
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⎝ 0 0 -1 ⎠ ⎝-1 0 0 ⎠ ⎝-1 0 0 ⎠
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The same is true for the S𝓍 operator in its basis. To express this operator in the z basis, however, it must be diagonalized.
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