Worked Problems in Physics

Nielsen and Chuang Exercise 2.60

Posted in Nielsen/Chuang by rpmuller on March 23, 2010

Exercise 2.60 Show that {(\vec v\cdot\vec\sigma)} has eigenvalues {\pm 1}, and that the projectors on to the corresponding eigenspace are given by {P_{\pm}=(I\pm\vec v\cdot\vec\sigma)/2}.

Starting from the spectral decomposition

\displaystyle U^\dagger(\vec v\cdot\vec\sigma)U=\Lambda

and squaring both sides yields

\displaystyle \Lambda^2=U^\dagger(\vec v\cdot\vec\sigma)UU^\dagger(\vec v\cdot\vec\sigma)U =U^\dagger(\vec v\cdot\vec\sigma)(\vec v\cdot\vec\sigma)U=U^\dagger IU=I.

(The second-to-last step uses the result from Exercise 2.35 that {(\vec v\cdot\vec\sigma)(\vec v\cdot\vec\sigma)=I}.) Since the eigenvalues of the Hermitian matrix {(\vec v\cdot\vec\sigma)} are real, {\lambda_i=\pm 1}.

We can easily verify that {P_{\pm}=(I\pm\vec v\cdot\vec\sigma)/2} meets the requirements for the project that {\sum_m P_m=I}, that {P_mP_n=\delta_{mn}P_m}, and that {M=\sum_m mP_m}. Thus, {P_{\pm}=(I\pm\vec v\cdot\vec\sigma)/2} is a set of projectors for {(\vec v\cdot\vec\sigma)}.