Show that the rotation matrices dm^' m^j(θ)=⟨j m^'|e^-i θJ2| j m⟩for j=(1)/(2) and j=1 are j=(1)

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Show that the rotation matrices dm^' m^j(θ)=⟨j m^'|e^-i θJ2|...

Show that the rotation matrices $$ d_{m^{\prime} m}^{j}(\theta)=\left\langle j m^{\prime}\left|e^{-i \theta J_{2}}\right| j m\right\rangle $$ for $j=\frac{1}{2}$ and $j=1$ are $$ j=\frac{1}{2}\left\{\begin{array}{l} d_{++}=d_{--}=\cos \frac{1}{2} \theta \\ d_{-+}=-d_{+-}=\sin \frac{1}{2} \theta \end{array}\right. $$ where $\pm$ denote $m=\pm \frac{1}{2}$, respectively, and $$ j=1\left\{\begin{array}{l} d_{01}=-d_{10}=-d_{0-1}=d_{-10}=\sqrt{\frac{1}{2}} \sin \theta \\ d_{11}=d_{-1-1}=\frac{1}{2}(1+\cos \theta) \\ d_{-11}=d_{1-1}=\frac{1}{2}(1-\cos \theta) \\ d_{00}=\cos \theta . \end{array}\right. $$

Show that the rotation matrices
$$
d_{m^{\prime} m}^{j}(\theta)=\left\langle j m^{\prime}\left|e^{-i \theta J_{2}}\right| j m\right\rangle
$$
for $j=\frac{1}{2}$ and $j=1$ are
$$
j=\frac{1}{2}\left\{\begin{array}{l}
d_{++}=d_{--}=\cos \frac{1}{2} \theta \\
d_{-+}=-d_{+-}=\sin \frac{1}{2} \theta
\end{array}\right.
$$
where $\pm$ denote $m=\pm \frac{1}{2}$, respectively, and
$$
j=1\left\{\begin{array}{l}
d_{01}=-d_{10}=-d_{0-1}=d_{-10}=\sqrt{\frac{1}{2}} \sin \theta \\
d_{11}=d_{-1-1}=\frac{1}{2}(1+\cos \theta) \\
d_{-11}=d_{1-1}=\frac{1}{2}(1-\cos \theta) \\
d_{00}=\cos \theta .
\end{array}\right.
$$

Key Concepts

Rotation Operators

Rotation operators in quantum mechanics represent the action of spatial rotations on quantum states. They are generated by angular momentum operators, and when exponentiated (as in exp(-i?J?)), they describe how states transform under a rotation by an angle ? about a specified axis. This concept is crucial for understanding how physical systems behave under symmetry transformations, which is central to many aspects of quantum theory.

Angular Momentum Operators

Angular momentum operators are the generators of rotations in quantum mechanics, obeying specific commutation relations that form the algebra of the rotation group. They are the quantum analogs of classical angular momentum and play a key role in determining the properties and transformation behavior of quantum states. Their eigenvalues label states in a given representation and help define the structure of rotation matrices.

Wigner d-matrix

The Wigner d-matrix elements, d????'?(?), are specific matrix elements of the rotation operator in the basis of angular momentum eigenstates. They encode the dependence of quantum state rotations on the angle ? for a given representation labeled by j. These matrices are fundamental in the study of rotational symmetries, especially in the manipulation and transformation of spin states in quantum systems.

Spin Representations

Spin representations describe the quantum mechanical property of intrinsic angular momentum, where the quantum number j can take half-integer values, such as 1/2 or 1. These representations are used to construct rotation matrices that act on spin states. Understanding spin representations is essential for analyzing systems like electrons in atoms or particles in quantum field theory, where half-integer and integer spins dictate the behavior under rotations.

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