Question

The operators P and Q commute and they are represented by the matrices |(1, 2) (2, 1)| and |(3, 2) (2, 3)| . Find the eigen vectors of P and Q. What do you notice about these eigen vectors, which verify a necessary condition for commuting operators?

Answer 1

Using the standard method explained in Chap. 1, define the eigen values λ₁ = 3 and λ₂ = −1 for the matrix P and the eigen vectors √ 1/√2|(1, 1)|  and 1/√2|(1, −1)|. For the matrix Q, the eigen values are λ₁ = 5 and λ₂ = 1, the eigen vectors being 1/√2|(1, 1)| and 1/√2|(1, −1)|. Thus the eigen vectors for the commutating matrices are identical.