Griffiths Introduction to Elementary Particles · Problem 4.6

Bound States — David J. Griffiths

Physics — QFT · rotation-matrix-so2

Problem

Consider a vector A\mathbf{A} in two dimensions. Suppose its components with respect to Cartesian axes xx, yy, are (ax,ay)(a_x, a_y). What are its components (ax′,ay′)(a'_x, a'_y) in a system x′x', y′y' which is rotated, counterclockwise, by an angle θ\theta, with respect to xx, yy? Express your answer in the form of a 2×22 \times 2 matrix R(θ)R(\theta): (ax′ay′)=R(axay)\begin{pmatrix} a'_x \\ a'_y \end{pmatrix} = R \begin{pmatrix} a_x \\ a_y \end{pmatrix} Show that RR is an orthogonal matrix. What is its determinant? The set of all such rotations constitutes a group; what is the name of this group? By multiplying the matrices, show that R(θ1)R(θ2)=R(θ1+θ2)R(\theta_1)R(\theta_2) = R(\theta_1 + \theta_2); is this an Abelian group?

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