Quantum Mechanics for Mathematicians

• For the three basis elementsljofso(3), show that the momentum map
  gives functionsμljthat are just the components of the angular momentum.


• Show that the maps
  lj∈so(3)→μlj
  give a Lie algebra homomorphism fromso(3) to the Lie algebra of functions
  on phase space (with Lie bracket on such functions the Poisson bracket).

Problem 3:
In the same context as problem 2, compute the Poisson brackets

{μlj,qk}

between the angular momentum functionsμljand the configuration space co-
ordinatesqj. Compare this calculation to the calculation of

π′(lj)ek

forπthe spin-1 representation ofSO(3) onR^3 (the vector representation).

Problem 4:
Consider the symplectic groupSp(2d,R) of linear transformations of phase
spaceR^2 dthat preserve Ω.

• Consider the group of linear transformations of phase spaceR^2 dthat act
  in the same way on positions and momenta, preserving the standard inner
  products on position and momentum space. Show that this group is a
  subgroup ofSp(2d,R), isomorphic toO(d).


• Using the identification betweenSp(2d,R) and matrices satisfying equa-
  tion 16.10, which matrices give the subgroup above?


• Again in terms of matrices, what is the Lie algebra of this subgroup?


• Identifying the Lie algebra ofSp(2d,R) with quadratic functions of the
  coordinates and momenta, which such quadratic functions are in the Lie
  algebra of theSO(d) subgroup?


• Consider the function
  1
  2

∑d

j=1

(q^2 j+p^2 j)

What matrix does this correspond to as an element of the Lie algebra of

Sp(2d,R)? Show that one gets anSO(2) subgroup ofSp(2d,R) by taking

exponentials of this matrix. Is thisSO(2) a subgroup of theSO(d) above?