Proof of vector identities using pauli matrices
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1. The problem statement, all variables and given/known data I'm supposed to derive the following:
( A ⋅ σ ) ( B ⋅ σ ) = A ⋅ B I + i ( A × B ) ⋅ σ (A⋅σ)(B⋅σ)=A⋅BI+i(A×B)⋅σ using just the two following facts:
Any 2x2 matrix can be written in a basis of spin matrices: M = ∑ m α σ α M=∑mασα which means that the beta-th component is given by m β = 1 2 T r ( M σ β ) mβ=12Tr(Mσβ) 2. Relevant equations listed above... 3. The attempt at a solution It should just be a left side= right side proof. I started by saying ( A ⋅ σ ) ( B ⋅ σ ) = ( ∑ α a α σ α ⋅ σ ) ( ∑ α b α σ α ⋅ σ ) = ⎛ ⎝ ∑ β a α δ α β ⎞ ⎠ ( ∑ γ b α δ α γ ) = ∑ β a β ∑ γ b γ = 1 2 T r ( A σ β ) 1 2 T r ( B σ γ ) (A⋅σ)(B⋅σ)=(∑αaασα⋅σ)(∑αbασα⋅σ)=(∑βaαδαβ)(∑γbαδαγ)=∑βaβ∑γbγ=12Tr(Aσβ)12Tr(Bσγ)
( A ⋅ σ ) ( B ⋅ σ ) = A ⋅ B I + i ( A × B ) ⋅ σ (A⋅σ)(B⋅σ)=A⋅BI+i(A×B)⋅σ using just the two following facts:
Any 2x2 matrix can be written in a basis of spin matrices: M = ∑ m α σ α M=∑mασα which means that the beta-th component is given by m β = 1 2 T r ( M σ β ) mβ=12Tr(Mσβ) 2. Relevant equations listed above... 3. The attempt at a solution It should just be a left side= right side proof. I started by saying ( A ⋅ σ ) ( B ⋅ σ ) = ( ∑ α a α σ α ⋅ σ ) ( ∑ α b α σ α ⋅ σ ) = ⎛ ⎝ ∑ β a α δ α β ⎞ ⎠ ( ∑ γ b α δ α γ ) = ∑ β a β ∑ γ b γ = 1 2 T r ( A σ β ) 1 2 T r ( B σ γ ) (A⋅σ)(B⋅σ)=(∑αaασα⋅σ)(∑αbασα⋅σ)=(∑βaαδαβ)(∑γbαδαγ)=∑βaβ∑γbγ=12Tr(Aσβ)12Tr(Bσγ)
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