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Simultaneous Diagonalization of 2 Hermitian Matrix.
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Simultaneous Diagonalization of 2 Hermitian Matrix • We earlier said that we if an operator has degenerate eigen values then the eigenket corresponding to the degenerate eigenvalue canonot define a unique direction in space. But takes you to a subspace, dimensionality of which is the order of the degeneracy. • Simultaneous diagonalization of two Hermitian operators is possible if and only if they commute. That is there exists atleast a basis which simultaneously diagonalises it. • Let us consider two operators, and which commute i.e. [, ]=0 • Let us consider the case where atleast one of the operators is non degenerate. We are assuming that has all non-degenerate eigenvalues • has degenerate eigenvalues. Hence its eigen kets cannot be used to define a unique basis. So we choose another operator but we are still interested in . राघववर्मा
Fixing a unique direction in Space राघववर्मा
Fixing a unique direction if both the operators are degenerate राघववर्मा
Complete Set of Commuting Operators राघववर्मा