Details

Quantum mechanics uses the commutator

as a definitive utility. Define angular momentum as the pseudo-vector quantity that satisfies

,

where is a scale factor with dimensions of action and is the Levi–Civita tensor of rank three. The commutator for the operators also defines the Lie algebra [2].

The algebra describes the gauge symmetry of the 2D quantum harmonic oscillator (QHO) and admits as a subalgebra, so it is possible to write the angular momentum operators in terms of the Pauli matrices and bilinear combinations of the creation/annihilation operators . Specifically,

,

with

,

, , .

The simple commutation relations

,

lead to a simple set of basis vectors, say , with , which are also the eigenfunctions of the operator

,

the Hamiltonian of the 2D QHO in frequency dimensions. After we know how the variables act on the basis vectors , it is possible to determine the matrix elements of any function in the basis provided by so long as the function can be expanded in a power series of the noncommuting variables. Fortunately, the operators have only quadratic terms, which makes explicit calculation of matrix elements easy.

The angular momentum operators have another, more natural set of quantum numbers: , . Each integer or half-integer indexes an irreducible representation of the angular momentum algebra. Then we construct the irreducible representations by taking the block diagonal subspace spanned by the basis vectors that also have quantum number . In the usual representation where is diagonal, the eigenvalues of representation range from to .

References

[1] W. Harter, "Principles of Symmetry, Dynamics, and Spectroscopy", Wiley, 1993. http://www.uark.edu/ua/modphys/markup/PSDS_Info.html

[2] F. Iachello, "Lie Algebras and Applications", Springer, 2014. http://link.springer.com/book/10.1007%2 F3-540-36239-8

[3] J. Schwinger, "On Angular Momentum," Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952. www.osti.gov/accomplishments/documents/fullText/ACC0111.pdf, www.ifi.unicamp.br/~cabrera/teaching/paper_schwinger.pdf.