Reflections on Quantum Mechanics for Electrical Engineers: From Kelvin's Telegraph Equation to the Static Charge Distribution

Quantum mechanics is often presented as a revolutionary break with classical physics—a mysterious theory comprehensible only through abstract postulates and philosophical interpretation. This paper shows how quantum mechanics can be understood not as a break, but as the logical continuation of familiar concepts. Starting from Lord Kelvin's 1855 telegraph equation, the paper demonstrates how the Schrödinger equation emerges as a special limiting case: a telegraph line with purely imaginary damping. The underlying mathematical structure—the Sturm-Liouville eigenvalue problem—was already worked out completely by Sturm and Liouville in 1836, ninety years before Schrödinger. Special functions (Hermite, Legendre, Laguerre polynomials) were developed for classical applications long before quantum mechanics. The Schrödinger equation's central task is the calculation of static charge distributions—|ψ(x)|²—which directly explains band structure, semiconductor physics, and all optoelectronic devices. Pure didactics: no new postulate, no new experiment, only textbook physics followed consistently to its end. Designed for electrical engineering students and educators seeking a unified, continuous approach to quantum mechanics rooted in classical transmission-line theory.