The Quantum Mechanical Problem of a Particle on a Ring with Delta Well

2022/11/29 by Berger, Raphael J. F.
#FOS: Physical sciences #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2211.16149

Abstract

The problem of a spin-free electron with mass m, charge e confined onto a ring of radius R₀ and with an attractive Dirac delta potential with scaling factor (depth) κ in non-relativistic theory has closed form analytical solutions. The single bound state function is of the form of a hyperbolic cosine that however contains a parameter d>0 which is the single positive real solution of the transcendental equation \coth(d) = λd for non zero real λ=(2)/(πκ). The energy eigenvalue of the bound state ε=-(d²)/(2π²)≈ (q e m R₀)/(2 ℏ²). In addition a discretly infinite set of unbounded solutions exists, formally these solutions are obtained from the terms for the bound solution by substituting d → i d yielding \cot(d) = λd as characteristic equation with the corresponding set of solutions d_k, k∈ℕ, the respective state functions can be obtained via \cosh(x)\oversetx → i x\longrightarrowcos(x).

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