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hermitian - Physics 225/315 Hermitian Operators Hermitian...

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Physics 225/315 January 25, 2008 Hermitian Operators Hermitian Adjoint An operator transforms a state. A | ψ = | ψ . The hermitian adjoint A transforms the corresponding dual state. ψ | = ψ | A . If A = A the operator or matrix is hermitian. The hermitian adjoint of a matrix is the complex conjugate transpose. Expectation value of a hermitian operator is real The relationship between a state and its dual is φ | ψ = ψ | φ * φ | A | ψ * = φ | ψ * = ψ | φ = ψ | A | φ Then ψ | A | φ = φ | A | ψ * If A = A then ψ | A | φ = φ | A | ψ * and if | φ = | ψ then ψ | A | ψ is real . This is the expectation value of A in the state | ψ . Eigenvalues of a hermitian operator are real A | ψ = λ | ψ Take inner product with ψ | then ψ | A | ψ = λ ψ | ψ The expectation value is real and ψ | ψ is real so λ is real. The spin operators S x , S y , S z are hermitian and all have eigenvalues ± ¯ h 2 . In the z representation the operators are: S x = ¯ h 2 ˆ 0 1 1 0 ! S y = ¯ h 2 ˆ 0 - i i 0 !
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