A Modern Approach to Quantum Mechanics · Problem 9.23

Bound States of Central Potentials — John S. Townsend

Physics — Quantum Mechanics · cylindrical-symmetry-hamiltonian

Problem

The Hamiltonian for a three-dimensional system with cylindrical symmetry is given by H^=p^22μ+V(ρ)\hat{H} = \frac{\hat{p}^2}{2\mu} + V(\rho) where ρ=x2+y2\rho = \sqrt{x^2 + y^2}. (a) Use symmetry arguments to establish that both p^z\hat{p}_z, the generator of translations in the zz direction, and L^z\hat{L}_z, the generator of rotations about the zz axis, commute with H^\hat{H}. (b) Use the fact that H^\hat{H}, p^z\hat{p}_z, and L^z\hat{L}_z have eigenstates in common to express the position-space eigenfunctions of the Hamiltonian in terms of those of p^z\hat{p}_z and L^z\hat{L}_z. Suggestion: Follow a strategy similar to the one that we followed in (9.94) for a spherically symmetric potential except that here we are using the eigenfunctions of p^z\hat{p}_z and L^z\hat{L}_z instead of the eigenfunctions L^2\hat{L}^2 and L^z\hat{L}_z. (c) What is the radial equation? Note: The Laplacian in cylindrical coordinates is given by ∇2ψ=1ρ∂∂ρ(ρ∂ψ∂ρ)+1ρ2∂2ψ∂ϕ2+∂2ψ∂z2\nabla^2\psi = \frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial\psi}{\partial\rho}\right) + \frac{1}{\rho^2}\frac{\partial^2\psi}{\partial\phi^2} + \frac{\partial^2\psi}{\partial z^2}

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