Quantum Field Theory I - Homework 9 | PHY 389K, Assignments of Quantum Mechanics

Download Quantum Field Theory I - Homework 9 | PHY 389K and more Assignments Quantum Mechanics in PDF only on Docsity! 2 Phug 381K [2406 4 Hw ¥ Realy ascrqu ment ; Sects Vit y id (2 P 167), Sect. 2.3 (ap 18 p. 172). 1. The deuteron ig a physical system composed of a proton with angular momentum j, = Jproton = 4 and a neutron with angular momentum j= jrewron = ¥4. a) Considering only rotational degrees of freedom, what is the space of physical states of the deuteron? b) Give the explicit expressions for the eigenvectors of deuteron angular momentum J in terms of the product basis vectors of jproton ANG freuwon- ¢) Calculate the expectation value of the operator J” - J’ pon - Fnewron for all the deuteron states. - Consider the combination of two rotators with * = 4 and j= 4. Start with the direct product basis vectors [J" m") @ LP m®) = |'4, 4) @ |'4, ys |94, 4) @ [4, “14s [4) @ [4 A): |4, 14 @ IA, 4). a) b) e) d) Gy 4, Which of these vectors are eigenvectors of J? = (J + 9 Form linear combinations of the remaining direct product vectors which are not eigenvectors of J to obtain normalized eigenvectors of J’. Check whether these are orthogonal to each other and to the vectors found in a. Calculate the action of Jz = J iJ, on the eigenvectors of F. Compare the coefficients that you found in b) with C G coefficients (44 m* ‘4m? | % 14 7 m) in Table 2.1, Sect. V.2 of the textbook. To evaluate the usefulness of the Wigner-Eckart theorem compare the number of matrix elements (j'm'|V2""|j m) with the number of reduced matrix elements (7" |] V? || 7) for a vector V. Show that the reduced matrix element for the angular momentum operator J, is given by GU FM} = 8590 +1) when the reduced matrix element is normalized as in (3.6). Compare this normaliza- tion with the normalization in your other favorite quantum mechanics books.