Lect4_BasisSet - Hermitian Operators Definition: an...

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Hermitian Operators Definition: an operator is said to be Hermitian if it satisfies: A =A Alternatively called ‘self adjoint’ In QM we will see that all observable properties must be represented by Hermitian operators Theorem: all eigenvalues of a Hermitian operator are real Proof: Start from Eigenvalue Eq.: Take the H.c. (of both sides): Use A =A: Combine to give: Since ! a m | a m " # 0 it follows that m m m a a A a ! = m m m a a A a ! = m m m m m m m m a a a a a a a A a = = ! m m a a = ! m m m a a a A = Eigenvectors of a Hermitian operator Note: all eigenvectors are defined only up to a multiplicative c-number constant Thus we can choose the normalization ! a m | a m " =1 THEOREM: all eigenvectors corresponding to distinct eigenvalues are orthogonal Proof: Start from eigenvalue equation: Take H.c. with m $ n: Combine to give: This can be written as: So either a m = a n in which case they are not distinct, or ! a m | a n " =0 , which means the eigenvectors are orthogonal ( ) ( ) m m m a c a a c A = ! m m m a a a A = m m m a a a A = n n n a a A a = m n m m n n m n a a a a a a a A a = = 0 ) ( = ! m n m n a a a a
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Completeness of Eigenvectors of a Hermitian operator THEOREM: If an operator in an M-dimensional Hilbert space has M distinct eigenvalues (i.e. no degeneracy), then its eigenvectors form a `complete set’ of unit vectors (i.e a complete ‘basis’) Proof: M orthonormal vectors must span an M-dimensional space. Thus we can use them to form a representation of the identity operator: Degeneracy Definition: If there are at least two linearly independent eigenvectors associated with the same eigenvalue, then the eigenvalue is
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Lect4_BasisSet - Hermitian Operators Definition: an...

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