7 Rotations and the Spin 1/2 Particle in a Magnetic Field

The existence of a non-trivial double cover $Spin(3)$ of the three dimensional rotation group may seem to be a somewhat obscure mathematical fact. Remarkably though, the existence of fundamental spin $\frac{1}{2}$ particles shows that it is $Spin(3)$ rather than $SO(3)$ that is the symmetry group corresponding to rotations of fundamental quantum systems. Ignoring the degrees of freedom describing their motion in space, which we will examine in later chapters, states of elementary particles such as the electron are described by a state space $\mathcal{H}={\mathbf{C}}^2$ , with rotations acting on this space by the two dimensional irreducible representation of $SU(2)=Spin(3)$ .

This is the same two-state system studied in chapter 3, with the $SU(2)$ action found there now acquiring an interpretation as corresponding to the double cover of the group of rotations of physical space. In this chapter we will revisit that example, emphasizing the relation to rotations.

中文目录

Chapter contents

• 7.1 The spinor representation
• 7.2 The spin 1/2 particle in a magnetic field
• 7.3 The Heisenberg picture
• 7.4 Complex projective space
• 7.5 The Bloch sphere
• 7.6 For further reading

Previous chapter: 6 The Rotation and Spin Groups in 3 and 4 Dimensions

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来源版本：2025-10-20

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