wang
(Wang)
#1
whereεijkis totally antisymmetric ini,j,kand is normalized toε 123 = 1.
We now study all possible quantum systems on which the angular momentum algebra
can be realized. Since the rotations generate symmetries of a quantum system, they must be
realized on the quantum system in terms of unitary operators. Theinfinitesimal generators
of these unitary transformations are precisely the angular momentum operatorsJ 1 ,J 2 ,J 3
which must be self-adjoint operators, i.e. observables. The commutation relations of theJ′s
indeed respect their self-adjointness.
5.7.1 Complete set of commuting observables
The angular momentum algebra possesses an operator (other than 0 orI) which commutes
with all threeJi, i= 1, 2 ,3, namely the Casimir operator
J^2 =J 12 +J 22 +J 32 (5.87)
Any other operator which commutes with all three Ji, i = 1, 2 ,3 must be functionally
dependent onJ^2. In this sense,J^2 is unique. Given the self-adjointness of theJi, i= 1, 2 ,3,
it is manifest thatJ^2 is also self-adjoint, and thus an observable.
To construct the Hilbert spaceHof a quantum system,we identify a complete set of
commuting observables, and then simultaneously diagonalize these observables to find a basis
forH. Assuming that the Hilbert spaceHcontains one and only one state associated with
each set of quantum numbers leads to an “irreducible quantum system”. Mathematically, this
means that we will find an “irreducible representation” of the angular momentum algebra. A
general quantum system associated with the angular momentum algebra may be reducible,
i.e. it can be decomposed as a direct sum of a number of irreducible systems.
We choose one of the generators, sayJ 3 as the first observable. Clearly, no non-zero linear
combination ofJ 1 andJ 2 commutes withJ 3 , but the Casimir operatorJ^2 does commute with
J 3. This is a standard result that a complete basis for all angular momentum states may
be parametrized by the quantum numbers ofJ 3 andJ^2. We shall label the states by|j,m〉
wherej,m∈R, since they correspond to eigenvalues of observables.
J^2 |j,m〉 = λ(j) ̄h^2 |j,m〉
J 3 |j,m〉 = m ̄h|j,m〉 (5.88)
The factors of ̄hhave been pulled out for later convenience.
5.7.2 Lowering and raising operators
We introducelowering and raising operatorsas follows,
J±=J 1 ±iJ 2 (5.89)
