A Modern Approach to Quantum Mechanics · Problem 2.9

Angular Momentum — John S. Townsend

Physics — Quantum Mechanics · adjoint-of-product-operators

Problem

Suppose in a two-dimensional basis that the operators

\hat{A}

and

\hat{B}

are represented by the

2\times 2

matrices

\hat{A} \to \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \qquad \hat{B} \to \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}

Show that

(\hat{A}\hat{B})^\dagger = \hat{B}^\dagger\hat{A}^\dagger