Principles of Quantum Mechanics · Problem 12.6.9

The Hydrogen Atom — R. Shankar

Physics — Quantum Mechanics · spherical-well-bound-states

Problem

Show that the quantization condition for l=0l = 0 bound states in a spherical well of depth −V0-V_0 and radius r0r_0 is k′/κ=−tan⁡k′r0k'/\kappa = -\tan k'r_0 where k′k' is the wave number inside the well and iκi\kappa is the complex wave number for the exponential tail outside. Show that there are no bound states for V0<π2ℏ2/8μr02V_0 < \pi^2 \hbar^2 / 8\mu r_0^2. (Recall Exercise 5.2.6.)

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