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Past exam of the advanced quantum physics course of the University of Cambridge / 2017 / 1 / b / Solution by

Codex  0 2026-10-18

The operators associated with the two angular momenta commute with one another, and each squared angular momentum operator commutes with every component of that same angular momentum. Two useful identities are

J_{z} = J_{1 z} + J_{2 z}, J_{1} \cdot J_{2} = \frac{1}{2} (J^{2} - J_{1}^{2} - J_{2}^{2}) .

(1)

Consequently the two maximal commuting subsets of the displayed collection are

{J_{1}^{2}, J_{2}^{2}, J_{1 z}, J_{2 z}, J_{z}}, {J_{1}^{2}, J_{2}^{2}, J^{2}, J_{z}, J_{1} \cdot J_{2}} .

(2)

The first set gives the uncoupled eigenbasis

∣ j_{1} m_{1} ⟩ \otimes ∣ j_{2} m_{2} ⟩

. Its

J_{z}

eigenvalue is already fixed by

m = m_{1} + m_{2}

. The second gives the coupled eigenbasis

∣ j_{1} j_{2}; jm ⟩

for the addition of angular momentum. Its scalar-product eigenvalue is already fixed by

j, j_{1}, j_{2}

. Thus an independent complete set of commuting observables uses four operators in either case; literal maximality within the question's list includes the dependent fifth operator.

For generic nonzero angular momenta, no further listed operator can be added. In particular,

[J_{1 z}, J_{1} \cdot J_{2}] = i ℏ (J_{1 y} J_{2 x} - J_{1 x} J_{2 y})

(3)

is generally nonzero, and

[J_{1 z}, J^{2}] = 2 [J_{1 z}, J_{1} \cdot J_{2}]

. The same obstruction applies to

J_{2 z}

. Their sum

J_{z}

commutes because the two commutators cancel.

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