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Unformatted text preview: Physics 225/315 January 25, 2008 Hermitian Operators Hermitian Adjoint An operator transforms a state. A | = | . The hermitian adjoint A transforms the corresponding dual state. | = | 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 | = | * | A | * = | * = | = | A | Then | A | = | A | * If A = A then | A | = | A | * and if | = | then | A | is real . This is the expectation value of A in the state | . Eigenvalues of a hermitian operator are real A | = | Take inner product with | then | A | = | The expectation value is real and | is real so is real. The spin operators S x , S y , S z are hermitian and all have eigenvalues h 2 ....
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- Spring '10