Consider the hydrogen atom: an electron of mass and charge 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 atom: an electron of mass mem_e and charge −e-e bound to a fixed (infinitely heavy) proton of charge +e+e by the Coulomb potential V(r)=−k/rV(r) = -k/r with k=e2/4πϵ0k = e^2/4\pi\epsilon_0. Define the classical Runge-Lenz vector M⃗=1mep⃗×L⃗−kr^\vec M = \frac{1}{m_e}\vec p\times\vec L - k\hat r (with r^=r⃗/r\hat r = \vec r/r), its Hermitian quantum counterpart 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}, and, on the subspace of bound (negative-energy) eigenstates of H^\hat H, the rescaled operator K⃗^=−me/2H^ M⃗^\hat{\vec K} = \sqrt{-m_e/2\hat H}\,\hat{\vec M}. You may use without re-deriving the two facts 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) Starting from Newton's second law p⃗˙=−kr^/r2\dot{\vec p} = -k\hat r/r^2 and the fact that a central force exerts no torque, show directly that dM⃗/dt=0d\vec M/dt = 0 for the classical orbit, and say what this implies for the orbit's shape. (b) Show that p⃗^×L⃗^\hat{\vec p}\times\hat{\vec L} alone is not Hermitian, that the symmetrised combination above is, and state precisely on what subspace K⃗^\hat{\vec K} is a well-defined Hermitian operator and why it fails elsewhere. (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 that A⃗^+\hat{\vec A}_+ and A⃗^−\hat{\vec A}_- each obey an angular-momentum algebra and that [A^+i,A^−j]=0[\hat A_{+i},\hat A_{-j}] = 0 for all i,ji,j. (d) Using A^+2=A^−2≡a(a+1)ℏ2\hat A_+^2 = \hat A_-^2 \equiv a(a+1)\hbar^2, the given identity, and n≡2a+1n \equiv 2a+1, derive EnE_n and the degeneracy of level nn. Evaluate E3E_3 numerically and state the degeneracy.

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