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



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.


[1] J. Schwinger, "On Angular Momentum," U. S. Atomic Energy Commission Report NYO-3071, January 26, 1952.

[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."

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