Homework 4 Questions for Quantum Mechanics I | PHY 389K, Assignments of Quantum Mechanics

Download Homework 4 Questions for Quantum Mechanics I | PHY 389K and more Assignments Quantum Mechanics in PDF only on Docsity! P389K HW4 1. Calculate the matrix elements of the angular momentum operators Ji, i = 1, 2, 3 in the representation space Rj= 1 2 and compare them with the Pauli matrices. 2. Find the matrix elements 〈f j ′ m′ |Si|f j m〉 of the three operators Si, i = 1, 2, 3 which fulfill the commmutation relation [Si, Sj ] = iijkSk and the anti-communtation relation {Si, Sj} ≡ SiSj + SjSi = 1 2 δij1 3. Calculate the matrices 〈j = l,m|Ji|j = l,m′〉 for j = 1, m, m′ = 1, 0,−1; i = 1, 2, 3. 5. Construct normalized eigenvectors∣∣∣j = 1 2 ,m1 〉 = α ∣∣∣j = 1 2 ,m = 1 2 〉 + β ∣∣∣j = 1 2 ,m = −1 2 〉 such that the |j, m1〉 are simultaneous eigenvectors of ~J 2 and J1, satisfying ~J 2|j, m1〉 = j(j + 1)|j, m1〉 and J1|j, m1〉 = m1|j, m1〉 Calculate the allowed values of m1. 6. Consider electromagnetic dipole decays of a rotating diatomic molecule. a.) Show that the frequency νj+1→j of the photon emitted in such a decay is given by νj+1→j = h 4π2I (j + 1) b.) The experimental values for νj+1→j are tabulated below for HCl in the far infrared j 4 5 6 7 8 9 10 11 ν[cm1] 83.03 104.13 124.37 145.37 165.89 186.23 206.60 288.86 1