# Introduction to quantum field theory/Spin

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## Exercise AB4

10:47, 1 November 2017 (UTC)

1. For any two three-dimensional vectors $\mathbf{A}$ and $\mathbf{B}$, prove that
$(\boldsymbol{\sigma} \cdot \mathbf{A})(\boldsymbol{\sigma} \cdot \mathbf{B}) = (\mathbf{A} \cdot \mathbf{B}) + \imath \boldsymbol\sigma \cdot (\mathbf{A} \times \mathbf{B})\,.$
2. Derive the $S_{z}, S_{x},\text{ and } S_{y}$ matrices for the case of a spin 1 particle.
3. Wavefunctions for a particle with some intrinsic property (such as spin) which can assume a fixed, finite number of components are often expressed as row vectors of functions of space (also called fields). Thus, wavefunctions for a spin-½ particle may be written as
$\Psi(\mathbf{x}) = \psi_{+}(\mathbf{x})\left\vert+\right\rangle + \psi_{-}(\mathbf{x})\left\vert-\right\rangle = \psi_{+}(\mathbf{x})\begin{pmatrix} 1 \\ 0 \end{pmatrix} + \psi_{-}(\mathbf{x})\begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} \psi_{+}(\mathbf{x}) \\ \psi_{-}(\mathbf{x}) \end{pmatrix}.$
The most general rotation operation that effects a spatial rotation and a corresponding rotation of the spin components is defined as $R_{\hat{\mathbf{n}}}(\theta) = e^{i\theta\hat{\mathbf{n}}\cdot\hat{\mathbf{J}}}\,,$ where $\hat{\mathbf{n}}$ represents a unit vector along some physical direction. Rotations by 2π affect the angular momentum eigenstates as follows: $R(2\pi) = (-1)^{2j}\left\vert j,m \right\rangle\,,$ irrespective of the direction $\hat{\mathbf{n}}$.
1. Show that for a particle of spin-0, rotation by an infinitesimal angle ε about the z-axis leads to the identification of $J_{z} = -i\hbar (x\partial/\partial y - y \partial/\partial x) \equiv L_{z}\,.$(Only to the first order in ε.)
2. Show that the matrix corresponding to the rotation operator in the spin state along some direction $\hat{\mathbf{n}} = (\sin\theta \cos\varphi, \sin\theta \sin\varphi, \cos\theta)$ is given by $\hat{\mathbf{n}}\cdot\mathbf{S} = \tfrac{1}{2}\hbar\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\,,$ where
$\hat{\mathbf{n}}.\boldsymbol{\sigma} = \begin{pmatrix} \cos\theta & e^{-\imath\varphi}\sin\theta \\ e^{\imath\varphi}\sin\theta & -\cos\theta \end{pmatrix} \,.$
4. Given two different particles in angular momentum states $\left\vert j_{1}, m_{1} \right\rangle$, and $\left\vert j_{2}, m_{2} \right\rangle$, we define $\left\vert j_{1}, j_{2}, m_{1}, m_{2} \right\rangle \equiv \left\vert j_{1}, m_{1}\right\rangle \left\vert j_{2}, m_{2} \right\rangle$. If the two particles interact to produce a composite particle in the angular momentum state $\left\vert J, M \right\rangle \equiv \left\vert j_{1}, j_{2}, J, M \right\rangle$, then one can write, using the completeness relation $\sum_{m}\left\vert j,m \right\rangle \left\langle j,m \right\vert = \mathbb{I}$ for any fixed $j$
$\left\vert j_{1}, j_{2}, J, M \right\rangle = \sum_{m_{1},m_{2}}\left\vert j_{1}, j_{2}, m_{1}, m_{2} \right\rangle \left\langle j_{1}, j_{2}, m_{1}, m_{2} \vert j_{1}, j_{2}, J, M \right\rangle.$
The coefficients $\left\langle j_{1}, j_{2}, m_{1}, m_{2} \vert j_{1}, j_{2}, J, M \right\rangle$ are called the Clebsch-Gordan coefficients, and are often written in a slightly shortened form: $\left\langle j_{1}, j_{2}, m_{1}, m_{2} \vert J, M \right\rangle.$
1. Show by considering rotations by 2π for the $\left\vert j_{1}, j_{2}, m_{1}, m_{2} \right\rangle$ and $\left\vert j_{1}, j_{2}, J, M \right\rangle$ states, that the coefficient $\left\langle j_{1}, j_{2}, m_{1}, m_{2} \vert J, M \right\rangle$ vanishes unless $j_{1} + j_{2} + J \in \mathbb{Z}\,.$
2. Argue that $\left\langle j_{1}, j_{2}, j_{1}, j_{2} \vert j_{1}, j_{2}, J, M \right\rangle = \delta_{J}^{j_{1}+j_{2}}\delta_{M}^{j_{1}+j_{2}}\,.$
3. What are the linearly independent states corresponding to $M = j_{1} + j_{2} - 1$? Calculate the corresponding Clebsch-Gordan coefficients.

Remember that the components of the spin angular momentum operators were defined along the same $x, y, \text{ and } z$ directions of the orbital angular momentum. So, a spatial rotation that alters these directions also affects the components of the spin angular momentum $S_{x}, S_{y},\text{ and } S_{z}.$