Franz Schwabl Quantum Mechanics 4th Edit · Problem 9.4

Addition of Angular Momenta

Physics — Quantum Mechanics · spinor-rotation-unitary-transformation

Problem

For a spatial rotation through angle φ\varphi about an axis of rotation along a unit vector n\boldsymbol{n}, transformations in spinor space are represented by U=eiφn⋅S/ℏ  .U = \mathrm{e}^{\mathrm{i}\varphi \boldsymbol{n} \cdot \boldsymbol{S}/\hbar} \;. (a) Show using (9.18b) that U=cos⁡φ/2+i(n⋅σ)sin⁡φ/2  .U = \cos\varphi/2 + \mathrm{i}(\boldsymbol{n} \cdot \boldsymbol{\sigma})\sin\varphi/2 \;. (b) Using this formula for UU and again exploiting the identities for the Pauli matrices, show that UσU†=n(n⋅σ)−n×[n×σ]cos⁡φ+[n×σ]sin⁡φ  .U\boldsymbol{\sigma} U^\dagger = \boldsymbol{n}(\boldsymbol{n} \cdot \boldsymbol{\sigma}) - \boldsymbol{n} \times [\boldsymbol{n} \times \boldsymbol{\sigma}]\cos\varphi + [\boldsymbol{n} \times \boldsymbol{\sigma}]\sin\varphi \;. (c) Discuss the special case n=ez\boldsymbol{n} = \boldsymbol{e}_z and consider infinitesimal rotations. (d) Finally, calculate for an arbitrary spinor (α+α−)\binom{\alpha_+}{\alpha_-} the transformed spinor defined by (α+′α−′)=U(α+α−)\begin{pmatrix} \alpha'_+ \\ \alpha'_- \end{pmatrix} = U \begin{pmatrix} \alpha_+ \\ \alpha_- \end{pmatrix} for n=ez\boldsymbol{n} = \boldsymbol{e}_z.

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