Consider a system, which consists of two distinguishable particles in 3-dimensio
ID: 2078007 • Letter: C
Question
Consider a system, which consists of two distinguishable particles in 3-dimensional space. Let's assume that the time evolution of the system is governed by the following Hamiltonian H, (a) H = H_0 (b) H = H_0 + H_1 (c) H = H_0 + H_1 + H_2 (d) H = H_0 + H_1 + H_2 + H_3 where H_0 = P_1^2/2m_1 + P_2^2/2m_2 + V_0 (r) H_1 = S_1 middot S_2 V_1 (r) H_2 = S middot LV_2 (r) H_3 = (S middot r)(s_2 middot r)V_3 (r). Here P_1, P_2, and R_1, R_2 are the linear momenta and the positions of the particles 1 and 2 with the mass m_1, m_2, respectively. And r = R_1 - R_2, r = |r|, L = L_1 + L_2, S = S_1 + S_2, J = L + S. For all Hamiltonians in (a, b, c, d), examine whether the following quantities are constants of motion (that is, conserved or not): L^2, L_z, S_z, J^2, J_z, and the parity under the inversion (R_1 rightarrow -R_1, R_2 rightarrow -R_2). Summarize your results in the table given below by marking "O" (if conserved) or "X" (if not conserved). [This problem will be graded based solely on the "O" or "X" marks in the table above. Each correct "O" or "X" mark is worth 1 point and there is NO partial credit. So you do not need to present your calculations.]Explanation / Answer
H= L2 LZ S2 SZ J2 Jz Parity H0 O O O X O X O H0+H1 X X O O X X X H0+H1+H2 O O O O O O X H0+H1+H2+H3 O O X X X X O