Consider an electron (with spin s = 1/2) in an l1 orbital of hydrogen (a) how many different values of me are there? how many different values of mg? how many total states |l, 8, me, ms) are there for this system? (b) recall that when adding angular momenta the possible values for the quantum number j range from l s to l+s in integer steps. what possible values of j exist for this system? for each value of j how many values of mj are there? how many total states |l, s; j, mj) are there? (c) draw a dot on the (j, m; ) plane (a graph where the horizontal axis is j and the vertical axis is mi) for each of the states you found in part (b). [note: each vertical column of dots in this graph represents a ladder for a fixed total angular momentum j! we know from problem 13.2(e) that the upper-right-most dot (the state with j = 3/2, mj is a "stretch state" and is equivalent to the ket |1,5,m +3/2) e = 1, mg =. (d) use the lowering operator j = l_+s_ to find the rest of the states 1, ; j= 3,mj) in terms of the basis e, s, me, ms). [hint: to start, we have |1, ; j= 3,mj = 3) = |1,3,mi apply j_to the left-hand side and l_ + s to the right-hand side. thus solve for the coupled basis ket 1,j, m; = )! rinse and repeat as needed. keep in mind that in the next step "down the ladder" you will now have two terms on the right-hand side to deal with.] 1,ms те 3 (e) check that your answers to part (d) agree with the expected results from the table of clebsch- gordan coefficients (see problem 13.4). ok, we've now generated the entire j = 3/2 ladder! but what about the two remaining states in the j = 1/2 ladder? (f) what two pairs (mt, ms) 1,3: ,m2)? 1,j, m+). use the fact possible in the expansion of the state are these same two states can also be combined to form the state that the j 3/2 and j = 1/2 states are made up only of the same two (me, ms) kets and must be orthogonal to find the expansion for the j 1/2 state [note: this only fixes the state up to an overall phase. we fix the phase by declaring that the coefficient for the ket with the largest value of me must be real and positivwe.] (g) act the lowering operator to find the remaining state.