Introduction to quantum field theory
Exercise AB4
10:47, 1 November 2017 (UTC)
- For any two three-dimensional vectors [math]\displaystyle{ \mathbf{A} }[/math] and [math]\displaystyle{ \mathbf{B} }[/math], prove that
[math]\displaystyle{ (\boldsymbol{\sigma} \cdot \mathbf{A})(\boldsymbol{\sigma} \cdot \mathbf{B}) = (\mathbf{A} \cdot \mathbf{B}) + \imath \boldsymbol\sigma \cdot (\mathbf{A} \times \mathbf{B})\,. }[/math]
- Derive the [math]\displaystyle{ S_{z}, S_{x},\text{ and } S_{y} }[/math] matrices for the case of a spin 1 particle.
- 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
[math]\displaystyle{ \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}. }[/math]
The most general rotation operation that effects a spatial rotation and a corresponding rotation of the spin components‡ is defined as [math]\displaystyle{ R_{\hat{\mathbf{n}}}(\theta) = e^{i\theta\hat{\mathbf{n}}\cdot\hat{\mathbf{J}}}\,, }[/math] where [math]\displaystyle{ \hat{\mathbf{n}} }[/math] represents a unit vector along some physical direction. Rotations by 2π affect the angular momentum eigenstates as follows: [math]\displaystyle{ R(2\pi) = (-1)^{2j}\left\vert j,m \right\rangle\,, }[/math] irrespective of the direction [math]\displaystyle{ \hat{\mathbf{n}} }[/math].
- Show that for a particle of spin-0, rotation by an infinitesimal angle ε about the z-axis leads to the identification of [math]\displaystyle{ J_{z} = -i\hbar (x\partial/\partial y - y \partial/\partial x) \equiv L_{z}\,. }[/math](Only to the first order in ε.)
- Show that the matrix corresponding to the rotation operator in the spin state along some direction [math]\displaystyle{ \hat{\mathbf{n}} = (\sin\theta \cos\varphi, \sin\theta \sin\varphi, \cos\theta) }[/math] is given by [math]\displaystyle{ \hat{\mathbf{n}}\cdot\mathbf{S} = \tfrac{1}{2}\hbar\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\,, }[/math] where
[math]\displaystyle{ \hat{\mathbf{n}}.\boldsymbol{\sigma} = \begin{pmatrix}
\cos\theta & e^{-\imath\varphi}\sin\theta \\
e^{\imath\varphi}\sin\theta & -\cos\theta
\end{pmatrix} \,. }[/math]
- Given two different particles in angular momentum states [math]\displaystyle{ \left\vert j_{1}, m_{1} \right\rangle }[/math], and [math]\displaystyle{ \left\vert j_{2}, m_{2} \right\rangle }[/math], we define [math]\displaystyle{ \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 }[/math]. If the two particles interact to produce a composite particle in the angular momentum state [math]\displaystyle{ \left\vert J, M \right\rangle \equiv \left\vert j_{1}, j_{2}, J, M \right\rangle }[/math], then one can write, using the completeness relation [math]\displaystyle{ \sum_{m}\left\vert j,m \right\rangle \left\langle j,m \right\vert = \mathbb{I} }[/math] for any fixed [math]\displaystyle{ j }[/math]
[math]\displaystyle{ \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. }[/math]
The coefficients [math]\displaystyle{ \left\langle j_{1}, j_{2}, m_{1}, m_{2} \vert j_{1}, j_{2}, J, M \right\rangle }[/math] are called the Clebsch-Gordan coefficients, and are often written in a slightly shortened form: [math]\displaystyle{ \left\langle j_{1}, j_{2}, m_{1}, m_{2} \vert J, M \right\rangle. }[/math]
- Show by considering rotations by 2π for the [math]\displaystyle{ \left\vert j_{1}, j_{2}, m_{1}, m_{2} \right\rangle }[/math] and [math]\displaystyle{ \left\vert j_{1}, j_{2}, J, M \right\rangle }[/math] states, that the coefficient [math]\displaystyle{ \left\langle j_{1}, j_{2}, m_{1}, m_{2} \vert J, M \right\rangle }[/math] vanishes unless [math]\displaystyle{ j_{1} + j_{2} + J \in \mathbb{Z}\,. }[/math]
- Argue that [math]\displaystyle{ \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}}\,. }[/math]
- What are the linearly independent states corresponding to [math]\displaystyle{ M = j_{1} + j_{2} - 1 }[/math]? Calculate the corresponding Clebsch-Gordan coefficients.
‡ Remember that the components of the spin angular momentum operators were defined along the same [math]\displaystyle{ x, y, \text{ and } z }[/math] directions of the orbital angular momentum. So, a spatial rotation that alters these directions also affects the components of the spin angular momentum [math]\displaystyle{ S_{x}, S_{y},\text{ and } S_{z}. }[/math]
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