Engineering Physics

4.5 Schrödinger’s time-independent wave equation

Based on de Broglie’s idea of matter waves, Schrödinger derived a mathematical equation known as Schrödinger’s wave equation. To derive Schrödinger’s wave equation, consider a particle of mass ‘m’ moving freely along X-direction [one dimensional] with velocity v. This moving particle is associated with a de Broglie wave of wavelength ‘λ’ and has frequency ‘v’. The expression for the displacement of a de Broglie wave associated with a moving particle is similar to an expression for undamped harmonic waves:

where ω = 2πν = angular frequency and v = νλ = velocity of the wave. ψ is called wave function, ...