Consider the hydrogen-like ion Li : a single electron of mass bound to a fixed…

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Radiative transitions, external fields and collisions

The Structure of Quantum Mechanics, Ch Hydrogen, and the symmetry that explains it

Problem

Consider the hydrogen-like ion Li2+^{2+}: a single electron of mass mem_e bound to a fixed (infinitely heavy) nucleus of charge +3e+3e, with Coulomb potential V(r)=−k/rV(r)=-k/r, k=3e2/4πϵ0k=3e^2/4\pi\epsilon_0. Define M⃗=1mep⃗×L⃗−kr^\vec M=\frac{1}{m_e}\vec p\times\vec L-k\hat r classically and, on the bound subspace, K⃗^=−me/2H^ M⃗^\hat{\vec K}=\sqrt{-m_e/2\hat H}\,\hat{\vec M} built from the Hermitian M⃗^=12me(p⃗^×L⃗^−L⃗^×p⃗^)−kr⃗^r.\hat{\vec M}=\frac{1}{2m_e}(\hat{\vec p}\times\hat{\vec L}-\hat{\vec L}\times\hat{\vec p}) - k\frac{\hat{\vec r}}{r}. You may use L⃗^⋅K⃗^=0\hat{\vec L}\cdot\hat{\vec K}=0 and K^2+L^2+ℏ2=−me2H^k2.\hat K^2+\hat L^2+\hbar^2 = -\frac{m_e}{2\hat H}k^2. Take hc=1239.84 eV nmhc=1239.84\,\text{eV\,nm}. (a) Show dM⃗/dt=0d\vec M/dt=0 classically for this potential and state the orbit consequence. (b) Show the symmetrised M⃗^\hat{\vec M} is Hermitian while the naive p⃗^×L⃗^\hat{\vec p}\times\hat{\vec L} is not, and state on which states K⃗^\hat{\vec K} is well defined. (c) Given [K^i,K^j]=iℏϵijkL^k[\hat K_i,\hat K_j]=i\hbar\epsilon_{ijk}\hat L_k and [L^i,K^j]=iℏϵijkK^k[\hat L_i,\hat K_j]=i\hbar\epsilon_{ijk}\hat K_k, form A⃗^±=12(L⃗^±K⃗^)\hat{\vec A}_\pm=\tfrac12(\hat{\vec L}\pm\hat{\vec K}) and show each obeys an angular-momentum algebra with [A^+i,A^−j]=0[\hat A_{+i},\hat A_{-j}]=0. (d) Derive EnE_n from the Casimir relation, and hence compute the wavelength of the photon emitted in the n=2→n=1n=2\to n=1 transition of Li2+^{2+} (the analogue of Lyman-alpha). State the degeneracy of the n=2n=2 level.

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