Question

5. (20 pts total) For a particle-on-a-ring (a simple model for the pi electrons in a cyclic aromatic molecule) the angular momentum operator L has eigenvectors n with eigenvalues Ln-nh, where n = 0, ±1, ±2, , too. L is a linear Hermitian operator representing a physically measurable quantity. The eigenvectors, ψ'm, are normalized. (The variable φ is the polar angle in 2D polar coordinates, with range 0 to 2, though this information should not be necessary.) a) (5 pts) In general, what values of angular momentum, L, are possible outcomes of an angular momentum measurement on this system? b) (5 pts) If the system is in state ЧЈ ei2ф, what is the probability, p(L), that any given value, L, of angular momentum will be found?

Answer 1

(a) L_n = n hstrok with n = 0, I_1, I_2--- are the possible out comes L^cap Psi_n = L_n psi_2 (b) p = 1 < psi_2 Vertical-bar psi_2 > Vertical-bar ^2 = Vertical-bar Integral^2 pi_0 e^-I 2 phi/squareroot 2 pi e^I n phi/squareroot 2 pi Vertical-bar ^2 = Vertical-bar 1/ 2 pi Integral^2 pi_0 e^i(n - 2) phi Vertical-bar ^2 = Vertical-bar 1/ 2 pi 2 pi delta_n,2 Vertical-bar ^2 = delta_n,2 so probability is 1 for n = 2 otherwise 0 (c).psi = 5/squareroot 2 g psi_2 + 2/ squareroot 2 g Psi_5 p_2 = Vertical-bar < Psi_2 Vertical-bar psi > Vertical-bar ^2 = Vertical-bar 5/ squareroot 29 Vertical-bar ^2 = 25/ 29 for n = 2 p_5 = Vertical-bar < psi_5 Vertical-bar psi > Vertical-bar = Vertical-bar 2/ squareroot 29 Vertical-bar = 4/29 for n = 5 & 0 fro all other n (d). < L > = 25/ 29 < L_2 > + 4/ 29 < L_5 > < L > = 25/29 (2 L) + h/ 29 times (5 hstrok) < L > = 70/29 hstrok