EXERCISE 2.4 Show that the four successive infinitesimal rotations ( εabout the 1-axis, followed

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EXERCISE 2.4 Show that the four successive infinitesimal...

EXERCISE $2.4$ Show that the four successive infinitesimal rotations ( $\varepsilon$ about the 1-axis, followed by $\eta$ about the 2 -axis, then $-\varepsilon$ about the 1-axis, and finally $-\eta$ about the 2 -axis) are equivalent to the second-order rotation $\varepsilon \eta$ about the 3 -axis. Hence, show that the generators satisfy $$ \left[J_{1}, J_{2}\right]=i J_{3} $$ Nonlinear functions of the generators which commute with all the generators are called invariants or Casimir operators. For the rotation group, $$ J^{2}=J_{1}^{2}+J_{2}^{2}+J_{3}^{2} $$ is the only Casimir operator, It follows that we can construct simultaneous eigenstates $|j m\rangle$ of $J^{2}$ and one of the generators, say $J_{3}$. Using only $(2.13)$, it is possible to show that $$ \begin{aligned} J^{2}|j m\rangle &=j(j+1)|j m\rangle \\ J_{3}|j m\rangle &=m|j m\rangle \end{aligned} $$ with $m=-j,-j+1, \ldots, j$, and where $j$ can take one of the values $0, \frac{1}{2}, 1, \frac{3}{2}, \ldots$.

EXERCISE $2.4$ Show that the four successive infinitesimal rotations ( $\varepsilon$ about the 1-axis, followed by $\eta$ about the 2 -axis, then $-\varepsilon$ about the 1-axis, and finally $-\eta$ about the 2 -axis) are equivalent to the second-order rotation $\varepsilon \eta$ about the 3 -axis. Hence, show that the generators satisfy
$$
\left[J_{1}, J_{2}\right]=i J_{3}
$$
Nonlinear functions of the generators which commute with all the generators are called invariants or Casimir operators. For the rotation group,
$$
J^{2}=J_{1}^{2}+J_{2}^{2}+J_{3}^{2}
$$
is the only Casimir operator,
It follows that we can construct simultaneous eigenstates $|j m\rangle$ of $J^{2}$ and one of the generators, say $J_{3}$. Using only $(2.13)$, it is possible to show that
$$
\begin{aligned}
J^{2}|j m\rangle &=j(j+1)|j m\rangle \\
J_{3}|j m\rangle &=m|j m\rangle
\end{aligned}
$$
with $m=-j,-j+1, \ldots, j$, and where $j$ can take one of the values $0, \frac{1}{2}, 1, \frac{3}{2}, \ldots$.

Key Concepts

Infinitesimal Rotations

Infinitesimal rotations are small angle rotations used to approximate and build finite rotations through a sequence of small steps. In the context of groups, their composition leads to a linearized approximation of the transformation, which is extremely useful in deriving the underlying algebraic structure of the rotation group, namely its Lie algebra.

Lie Algebra of Rotations

The generators of rotations, often denoted J1, J2, J3 in three dimensions, obey specific commutation relations that define the Lie algebra of the rotation group SO(3) or its covering group SU(2). An important example is the relation [J1, J2] = iJ3, which reveals the intrinsic non-commutativity of successive rotations and encodes the structure constant of the algebra.

Generators and Their Commutators

Generators are the fundamental elements that produce rotations when exponentiated. Their commutation relations provide a systematic way to capture the properties of the group. The calculation of successive infinitesimal rotations shows that their non-commutative behavior results in terms that are of second order in the infinitesimal parameters, leading directly to the structure of the commutators.

Casimir Operators

Casimir operators are invariant operators constructed from the generators that commute with all elements of the Lie algebra. For the rotation group, the quadratic Casimir operator J² = J1² + J2² + J3² is the unique invariant, which plays a crucial role in classifying the irreducible representations of the group in both classical and quantum contexts.

Angular Momentum Eigenstates

Angular momentum eigenstates, typically denoted |j m?, arise from the diagonalization of both the Casimir operator and one of the individual generators (commonly J3). These states are characterized by quantum numbers j and m, where j relates to the total angular momentum (with eigenvalue j(j+1)) and m represents the projection of this angular momentum, thereby providing a complete description of the state in the context of rotational symmetry.

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