Simultaneous Diagonalization of 2 Hermitian Matrix

1. 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

2. Fixing a unique direction in Space राघववर्मा

3. Fixing a unique direction if both the operators are degenerate राघववर्मा

4. Complete Set of Commuting Operators राघववर्मा