Schwinger's Oscillator Model for Angular Momentum

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In an internal Atomic Energy Commission document published in 1952 [1], Julian Schwinger developed the quantum theory of angular momentum from the commutation relations for a pair of independent harmonic oscillators. This work has since been subsequently quoted many times [2, 3]. A simplified version of the derivation is given in the Details. This Demonstration shows the angular momentum states ,
derived from the quantum numbers
for a pair of harmonic oscillators, with
.
Contributed by: S. M. Blinder (February 2020)
Open content licensed under CC BY-NC-SA
Snapshots
Details
A pair of uncoupled harmonic oscillators, designated a and b, can be defined by raising and lowering operators with commutation relations
,
.
It can then be shown that the operators
,
,
obey the canonical commutation relations for angular momentum:
,
,
,
.
The number operators for the two oscillators are given by
,
,
,
with corresponding eigenvalues ,
,
, each equal to an integer
.
In terms of the number operators, relevant angular momentum operators can be expressed as
,
.
The quantum number evidently can be identified with
, with possible values
. Analogously,
, running from
to
in integer steps.
References
[1] J. Schwinger, "On Angular Momentum," U. S. Atomic Energy Commission Report NYO-3071, January 26, 1952. www.osti.gov/biblio/4389568.
[2] J. J. Sakurai, Modern Quantum Mechanics, Menlo Park, CA: Benjamin/Cummings, 1985 pp. 217–221.
[3] H. Verma, T. Mitra and B. P. Mandal, "Schwinger's Model of Angular Momentum with GUP." arxiv.org/abs/1808.00766.
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