A particle is confined to the surface of a sphere of fixed radius

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Angular momentum from the rotation group

Problem

A particle is confined to the surface of a sphere of fixed radius, so its states are wavefunctions ψ(x,y,z)\psi(x,y,z) that, evaluated on the sphere, depend only on the direction (θ,ϕ)(\theta,\phi) of r⃗\vec r. Consider a rotation by a small angle ϵ\epsilon about the xx-axis, with R^(x^,ϵ)=1−iϵL^x/ℏ+O(ϵ2)\hat{\mathcal R}(\hat x,\epsilon)=\mathbb 1-i\epsilon\hat L_x/\hbar+O(\epsilon^2). ((a)) Show that, to first order in ϵ\epsilon, (R^(x^,ϵ)ψ)(x,y,z)=ψ(x, y+ϵz, z−ϵy)≈ψ(x,y,z)+ϵ(z∂ψ∂y−y∂ψ∂z)\big(\hat{\mathcal R}(\hat x,\epsilon)\psi\big)(x,y,z)=\psi(x,\,y+\epsilon z,\,z-\epsilon y)\approx \psi(x,y,z)+\epsilon\Big(z\dfrac{\partial\psi}{\partial y}-y\dfrac{\partial\psi}{\partial z}\Big), and match this to the definition of R^(x^,ϵ)\hat{\mathcal R}(\hat x,\epsilon) to derive L^x=y^p^z−z^p^y\hat L_x=\hat y\hat p_z-\hat z\hat p_y, the xx-component of r⃗^×p⃗^\hat{\vec r}\times\hat{\vec p}. ((b)) Every direction on the sphere is equivalent, so the same single-valuedness argument used for the zz-axis applies to the xx-axis: a 2π2\pi rotation about the xx-axis returns the sphere to itself. Use this to show that the eigenvalues of L^x/ℏ\hat L_x/\hbar must also be integers.

Related theory

Browse the problem library