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

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems

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

Problem

Consider the hydrogen-like ion He+^+: a single electron of mass mem_e bound to a fixed (infinitely heavy) nucleus of charge +2e+2e, with Coulomb potential V(r)=−k/rV(r) = -k/r, k=2e2/4πϵ0k = 2e^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 its bound-subspace quantum version 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 without re-deriving 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. (a) Using the Hamiltonian formulation (q˙i=∂H/∂pi\dot q_i = \partial H/\partial p_i, p˙i=−∂H/∂qi\dot p_i = -\partial H/\partial q_i, so f˙={f,H}PB\dot f = \{f,H\}_{\rm PB} for any phase-space function ff), show that {M⃗,H}PB=0\{\vec M,H\}_{\rm PB}=0, and state what this implies about the orbit. (b) Show that the naive p⃗^×L⃗^\hat{\vec p}\times\hat{\vec L} is not Hermitian while the symmetrised combination above is, and state exactly why K⃗^\hat{\vec K} is restricted to negative-energy eigenstates of H^\hat H. (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, define 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 as in the text's derivation of the spectrum, and evaluate the ground-state energy E1E_1 numerically. State the degeneracy of the n=2n=2 level and how it splits between l=0l=0 and l=1l=1.

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