The exact meaning of the angular-momentum and spin operators in quantum mechanics
Résumé
The theory of angular momentum and spin in quantum mechanics
seems to defy common-sense intuition.
We render the theory intelligible again by pointing out that this apparent
impenetrability merely stems from an {\em undue} parallel interpretation
of the algebraic expressions for the angular-momentum
and spin operators in
the group representation theory of SO(3) and SU(2). E.g. the correct meaning of
${\hat{L}}_{z} = {\hbar\over{\imath}}\,(x{\partial\over{\partial y}} - y{\partial\over{\partial x}} )$ is not that it is the operator
for the $z$-component $L_{z}$ of the angular momentum
${\mathbf{L}}$, but rather the expression of the operator
for the angular momentum ${\mathbf{L}}$ when it is aligned with the $z$-axis.
Hence what we are used to note (erroneously) as ${\hat{L}}_{z}$ is not a scalar but a vector operator.
The same applies
{\em mutatis mutandis} for the spin operators. In the correct interpretation, the whole
algebraic formalism is just the group representation theory for the rotations
of three-dimensional Euclidean geometry. It is thus mere,
elementary high-school mathematics (in a less usual, more technical guise) and as such totally exempt of any physics, let alone quantum mysteries.
The change of interpretation has no impact on the algebraic results, such that they remain in agreement with experimental data. It is all only a matter of the correct geometrical
meaning of the algebra.
All these statements are proved within the framework of the group representation theory
for SO(3) and SU(2)
which is the basic tool used to describe rotational motion
in quantum mechanics.
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