You are given a real operator A satisfying the quadratic equation ​

Question

You are given a real operator \(\hat A\) satisfying the quadratic equation

\(\hat A^2 - 3\hat A + 2 = 0\)

This is the lowest-order equation that \(\hat A\) obeys.

(a) What are the eigenvalues of \(\hat A\)?

(b) What are the eigenstates of \(\hat A\)?

(c) Prove that \(\hat A\) is an observable.

Answer 1

(a) As \(\hat A\) satisfies a quadratic equation it can be represented by a 2 x 2 matrix. Its eigenvalues are the roots of the quadratic equation

\(\lambda^2 - 3X + 2= 0, \lambda_1= 1, \lambda_2 = 2\)

(b) \(\hat A\) is represented by the matrix

then gives a = 1, b = 0 for \(\lambda \) = 1 and a = 0, b = 1 for \(\lambda \) = 2. Hence the eigenstates of \(\hat A\) are \(\begin{pmatrix}1\\0\end{pmatrix}\) and \(\begin{pmatrix}0\\1\end{pmatrix}\).

(c) Since \(\hat A\) = A⁺, \(\hat A\) is Hermitian and hence an observable.