Standing waves occur in many physical systems. The electron orbits in a hydrogen atom share important features with standing waves on a stretched string that is fixed at both ends.
Use this analogy to deepen your understanding of atomic energy quantization.
(a) Identify two specific similarities and one key difference between the allowed electron orbits in a hydrogen atom and the normal modes of a string of length L with fixed ends.
(b) For a stretched string of linear mass density μ under tension T, derive an expression for the allowed frequencies fn of its standing-wave modes. Express your result in terms of n,L,T, and μ.
(c) In the atomic model, increasing the nuclear charge from Z=1 (hydrogen) to Z=2 (singly-ionized helium) alters the Coulomb attraction while preserving the standing-wave condition. Using your reasoning from part (b), explain qualitatively how the change from Z=1 to Z=2 affects the spacing between adjacent energy levels.