Quantum Angular Momentum Matrices
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It is useful to have matrix representations of angular momentum operators for any quantum number . Matrix representations can be used, for example, to model the spectrum of a rotating molecule [1].
Contributed by: Brad Klee (January 2016)
Open content licensed under CC BY-NC-SA
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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.
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