Question

Tl 6. (4 pts) We calculated Ahf for the ground state Hydrogen atom, but now let us consider Hht for the ground state of the deuterium atom. The basic idea is the same: Here the matrix representation of S is the same, since it represents the spin operator for the spin- electron, but that of 7 is different because the deuteron is a spin-1 particle (neutron+proton) whose spin has 3 possible values for the magnetic quantum number. Also, we must use the reduced Ap - P A, because the gyromagnctic factor of the deuteron is smaller than that of the proton: gD -0.857. a. In the decoupled representation Imşmj), what are all the possible states? (Think of all possible combinations of spins.) b. In the coupled representation |FMF), what are all the possible states? (Note: (1) There should be the same number as in part a); (2) look at Table 11.3 for possible help.) c. Find the matrix representations of S2,12 and F2 in the coupled basis and use this to show that Hhf is diagonal in this basis. Comment on what the matrix element values of Hhf represent. Is the hyperfine splitting of ground state deuterium greater or less than that of hydrogen? Find the wavelength of the photon emitted from this splitting. Would this result imply that the hyperfine deuterium emission line is more or less useful for astronomy? Explain d. e. Table 11.3 Clebsch-Gordan Coefficients for j, 1 and j - m1 0 1 0 20 0 0 0 0 0 0 0

Answer 1

入, AD e2 수2-22-1 tip

2h2 并X- 2. b) F. o 1 oa No.ot sly:6

121 2 wじ54 be io.al wo ll diegon dlee

Li 1 22 33 2. 2 A2 S 522 Same cy 스2- 2. 31 131 2

lemont u1 be 1to E2 2 Ap 2- EAD

メEnegy ckFkrenG-E2-4-ADr..h lo COED 2D he (一 2- hc