Introduction to Quantum Mechanics · Problem 2.46

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · circular_ring

Problem

Imagine a bead of mass mm that slides frictionlessly around a circular wire ring of circumference LL. (This is just like a free particle, except that ψ(x+L)=ψ(x)\psi(x + L) = \psi(x).) Find the stationary states (with appropriate normalization) and the corresponding allowed energies. Note that there are (with one exception) two independent solutions for each energy EnE_n—corresponding to clockwise and counter-clockwise circulation; call them ψn+(x)\psi_n^+(x) and ψn−(x)\psi_n^-(x). How do you account for this degeneracy, in view of the theorem in Problem 2.44 (why does the theorem fail, in this case)?

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