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35 lines
847 B
Plaintext
35 lines
847 B
Plaintext
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Eigenvalue equations for the S𝓏 operator in a spin-1/2 system:
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S𝓏|+> = +ħ/2|+>
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S𝓏|-> = -ħ/2|->
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Matrix form of an operator:
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S𝓏 ≐ ( a b )
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( c d )
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Eigenvalue equations in matrix form:
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( a b )( 1 ) = +ħ/2 ( 1 )
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( c d )( 0 ) ( 0 )
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( a b )( 0 ) = -ħ/2 ( 0 )
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( c d )( 1 ) ( 1 )
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Can show with matrix multiplication operations that
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S𝓏 ≐ ħ/2 ( 1 0 )
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( 0 -1 )
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Some Properties:
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An operator is always diagonal in its own basis.
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Eigenvectors are unit vectirs in their own basis.
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S𝓏 ≐ ħ/2 ( 1 0 ) |+> = ( 1 ) |-> = ( 0 )
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( 0 -1 ) ( 0 ) ( 1 )
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Diagonalizing an Operator
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Diagonize a matrix --> find the eigenvalues and eigenvector
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S𝓍 ≐ ħ/2 ( 0 1 )
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( 1 0 )
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(1) 1⁄2 ⁵⁄ₐ
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