The Schrödinger wave equation is derived by combining the wave nature of particles with classical energy relationships. According to de Broglie, particles behave like waves with wavelength λ = h/p. For a particle of mass m moving in a potential V, the total energy is:

E = KE + V = p²/2m + V

By treating the particle as a wave and applying wave mathematics, momentum p is replaced with the operator −iħ∇-iħ\nabla−iħ∇ and energy E with iħ∂∂tiħ \frac{\partial}{\partial t}iħ∂t∂​. Substituting these into the classical energy equation gives the time-dependent Schrödinger equation:

iħ∂ψ∂t=−ħ22m∇2ψ+Vψiħ \frac{\partial ψ}{\partial t} = -\frac{ħ^2}{2m} \nabla^2 ψ + Vψiħ∂t∂ψ​=−2mħ2​∇2ψ+Vψ

This equation describes how the wave function ψ of a quantum particle behaves and forms the foundation of quantum mechanics.

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