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31 lines
507 B
Plaintext
31 lines
507 B
Plaintext
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S𝓍 ≐ ħ͟⎛0 1⎞
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2⎝1 0⎠
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Diagonalize the matrix...
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First, find the characteristic equation and solve for the eigenvalues.
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S𝓍 ≐ ħ͟⎛0 1⎞
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2⎝1 0⎠
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ħ͟ ⎛-λ 1⎞
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2 ⎝ 1 -λ⎠
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λ² - ħ²/4 = 0
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λ² = ħ²/4
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λ = ±ħ/2
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So the eigenvalues are those expected for the measurement of a spin-1/2 component.
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The eigenvalue equations are
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S𝓍❙Ψ❭ = λ❙Ψ❭
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which is represented by
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ħ͟⎛0 1⎞⎛a⎞ = λ⎛a⎞
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2⎝1 0⎠⎝b⎠ ⎝b⎠
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