Assume the operators Ä and B commute with each other, show that a) The matrix representation B in the basis |A¡), |A2), .. |AN) is a diagonal matrix b) The kets |A1), [A2), ... |An) are also eigenvectors of B c) The eigenvalues of B are given by (A¡|B|A¡)
Assume the operators Ä and B commute with each other, show that a) The matrix representation B in the basis |A¡), |A2), .. |AN) is a diagonal matrix b) The kets |A1), [A2), ... |An) are also eigenvectors of B c) The eigenvalues of B are given by (A¡|B|A¡)
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Matrix representation
Transcribed Image Text:Assume the operators Ä and B commute with each other, show that
a) The matrix representation B in the basis [A,), |A2),.. |AN) is a diagonal matrix
b) The kets |A1), |A2), ... |AN) are also eigenvectors of B
c) The eigenvalues of B are given by (A;|B|A;)
Consider that:
• la) is an arbitrary ket from the vector space V
• operators  and B are linear operators acting on vectors from V
• the set of all eigenvectors of  is given by |A1), |A2), ... |AN), and form an orthonormal basis
Aij and Bij are matrix elements of the matrix representations of the operators  and B
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