Introduction to Quantum Mechanics · Problem 6.32

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · spin_rotations

Problem

Rotations on spin states are given by an expression identical to Equation 6.32, with the spin angular momentum replacing the orbital angular momentum: Rn(φ)=exp⁡ ⁣[−iφℏ n⋅S].\mathsf{R}_{\mathbf{n}}(\varphi) = \exp\!\left[-i\frac{\varphi}{\hbar}\,\mathbf{n}\cdot\mathbf{S}\right]. In this problem we will consider rotations of a spin-1/21/2 state. (a) Show that (a⋅σ)(b⋅σ)=a⋅b+i (a×b)⋅σ,(\mathbf{a}\cdot\boldsymbol{\sigma})(\mathbf{b}\cdot\boldsymbol{\sigma}) = \mathbf{a}\cdot\mathbf{b} + i\,(\mathbf{a}\times\mathbf{b})\cdot\boldsymbol{\sigma}, where the σi\sigma_i are the Pauli spin matrices and a\mathbf{a} and b\mathbf{b} are ordinary vectors. Use the result of Problem 4.29. (b) Use your result from part (a) to show that exp⁡ ⁣[−iφℏ n⋅S]=cos⁡ ⁣(φ2)−i sin⁡ ⁣(φ2) n⋅σ.\exp\!\left[-i\frac{\varphi}{\hbar}\,\mathbf{n}\cdot\mathbf{S}\right] = \cos\!\left(\frac{\varphi}{2}\right) - i\,\sin\!\left(\frac{\varphi}{2}\right)\,\mathbf{n}\cdot\boldsymbol{\sigma}. Recall that S=(ℏ/2) σ\mathbf{S} = (\hbar/2)\,\boldsymbol{\sigma}. (c) Show that your result from part (b) becomes, in the standard basis of spin up and spin down along the zz axis, the matrix Rn=cos⁡ ⁣(φ2)(1001)−i sin⁡ ⁣(φ2)(cos⁡θsin⁡θ e−iϕsin⁡θ eiϕ−cos⁡θ)\mathsf{R}_{\mathbf{n}} = \cos\!\left(\frac{\varphi}{2}\right)\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix} - i\,\sin\!\left(\frac{\varphi}{2}\right)\begin{pmatrix}\cos\theta & \sin\theta\,e^{-i\phi}\\ \sin\theta\,e^{i\phi} & -\cos\theta\end{pmatrix} where θ\theta and ϕ\phi are the polar coordinates of the unit vector n\mathbf{n} that describes the axis of rotation. (d) Verify that the matrix Rn\mathsf{R}_{\mathbf{n}} in part (c) is unitary. (e) Compute explicitly the matrix Sx′=R† Sx R\mathsf{S}'_x = \mathsf{R}^\dagger\,\mathsf{S}_x\,\mathsf{R} where R\mathsf{R} is a rotation by an angle φ\varphi about the zz axis and verify that it returns the expected result. *Hint*: rewrite your result for Sx′\mathsf{S}'_x in terms of Sx\mathsf{S}_x and Sy\mathsf{S}_y. (f) Construct the matrix for a π\pi rotation about the xx axis and verify that it turns an up spin into a down spin. (g) Find the matrix describing a 2π2\pi rotation about the zz axis. Why is this answer surprising?

Related theory

Browse the problem library