Addition of two spin-1/2 angular momenta (30 points) for different eigenstates.) if no, explain why not and indicate what states would be eigenstates of ś consider a composite particle composed of a bound state of two spin-1/2 particles a and b. as examples, this could be the electron and a proton in a (d). once we know the m quantum nuber of each state, we would also like to hydrogen atom, or two electrons paired in a superconducting state (a cooper know the total spin quantum number s (the spin analog of the orbital angular pair). here we ignore the orbital angular momentum, and concentrate just on momentum e.) to do this, we will need the operator s2, which may be written the spin angular momentum. (in the real world, one has to worry about the as orbital angular momentum as well, but we will get to that next semester.) (22) (a). to begin with, assume that the two particles have general spin values sa and sg. each particle has its own spin operators: sa, s4, and sa, for particle a, and s, s, for particle b, as well as separate spin quantum numbers, which we may label as sa and ma for particle a, and sp and mo for particle b. including only spin, the basis vectors may be written as first, prove that sa·sb can be expressed as (23) (18) where the raising and lowering operators for cach particle are defined in the usual way (note that a single state must identify a spin configuration for both particles) the spin operators for particle a and particle b are independent. given this, now take the state defined with the largest m value obtained in part (b write down the result of the operations of sa 2。 (sb 2 s and s on a general and operate on it with 2 what is the value of two-spin state of the form sa, mai sb, mb). keep sa and sb fully general in have predicted tis from the fact that this was the largest value of a/? this part for this state? could vou (e how many states are there for a combined particle of the spin s you (b). now let the the particles both have spin-1/2, so that sa sb/2found in part (d)? act on the state with highest m with the lowering operator because these numbers will never change, it is "boring to write them over and s_ sa+sb, and use eq. (7), substituting for s and m either s and m, sa and over again, and so we may just write the spin states of the paired particles as ma, sp andmb, each as appropriate. what are the s and m values of the state you obtained, and what is this state in terms of the original states from part (b)? (19) repeat this process until you have run out of states for that value of spin s how many different states are there for two spin-1/2 particles? list them all. (hint: there aren't too many more.) if you like, you may use the standard notation of using the uparrow (t) and downarrow ( symbols to denote m 1/2 and m1/2, respectively, so that (f). do the states you found in part (e) span the entire hilbert space? if not, how many states are missing? explain what this or these missing state(s) are in terms of the original states from part (b). what value or values of m does it (20) or do they have? construct it or them to be orthogonal to al of the states with spin s, normalized, and if necessary, orthonormal to one another. assume that . all of these remaining state(s) fit into a single other spin value s'. what value (c). when particles a and b are combined into a composite particle, it is often must s have? to find t explicitly act on the missing state with the highest useful to ask about the total spin of the composite. in particular, interactions m value with the operator st as in part (e). then act on it with s as many we will take the ful set of allowed states to form the basis for our hilbert space times as are necessary to complete the hilbert space with other particles and fields will often depend upon the total spin, so we would like to determine what that is. define the total spin vector operator as (21) are the states you listed in part (b) eigenstates of s? if yes, give the value am for the quantum number associated with s for each of your eigenstates, and give a general relation between m and ma and mb. (note: m can be different