advertisement

PHY 6646 - Quantum Mechanics II - Spring 2011 Homework set # 3, due January 26 1. Show that √ ⎛ Σ(1) x 0 1⎜√ = ⎝ 2 2 0 0 1⎜ √ = ⎝i 2 2 0 ⎛ Σ(1) z = 1 ⎜ ⎝0 0 √0 ⎟ 2⎠ 0 √ −i 2 0 √ i 2 ⎛ Σ(1) y ⎞ 2 0 √ 2 0 0 0 ⎞ 0√ ⎟ −i 2⎠ 0 ⎞ 0 0 ⎟ ⎠ −1 (0.1) and ⎛ Σ(1) x 0 ⎜ =⎝0 0 ⎛ Σ(1) y 0 ⎜ =⎝ 0 −i 0 0 0 0 ⎜ = ⎝ +i 0 −i 0 0 ⎛ Σ(1) z 0 0 +i ⎞ 0 −i⎟ ⎠ 0 ⎞ +i 0 ⎟ ⎠ 0 ⎞ 0 0⎟ ⎠ 0 (0.2) are equivalent representations of the rotation group SO(3), i.e. that there exists a unitary † (1) † (1) † matrix U such that Σ(1) = UΣ(1) = UΣ(1) = UΣ(1) x x U , Σy y U and Σz z U . What is the matrix U? 2. Problems 15.2.1 (both parts), 15.2.2(2) (second part only), and 15.2.5 in Shankar’s book. (Note: There is a typo in the statement of problem 15.2.5, which you have to correct as part of your answer to the question). 1