Quantum physics is based on wave-particle duality, which for brevity I will express as each particle has an associated wave representation.

In 1600s, Newton gave us E = p²/2m, which connects the energy of a particle with its momentum and mass.

In 1700s, Euler gave us the wave equation Ψ = e^{i(kx-ωt)}, which describes a wave with time frequency ω and space frequency k.

In 1899, Plank makes THE quantum hypothesis E = hν (or E = ħω) which corresponds the energy of a particle to the time frequency of a wave.

In 1924, de Broglie combines E=hν & E=pV=pνλ to get p = h/λ (or p = ħk) which relates the momentum of a particle to the space frequency of a wave.

In 1926, Schrödinger, used the preceding, to derive the quantum wave equation:

dΨ/dt = -iωe^{i(kx-ωt)} = -iωΨ = -i(E/ħ)Ψ

iħd/dt Ψ = EΨ

dΨ/dx = ike^{i(kx-ωt)} = ikΨ = i(p/ħ)Ψ

pΨ = -iħd/dx Ψ

p²Ψ = -ħ²d²/dx² Ψ

p²/2m Ψ = -ħ²/2m d²/dx² Ψ

iħd/dt Ψ = EΨ = p²/2m Ψ = -ħ²/2m d²/dx² Ψ

**iħ$\frac{d}{\mathrm{dt}}$ Ψ = -**

This is the Schrödinger equation.