Wave Packet

PHY308 : Quantum Physics

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 = ei(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 = -iωei(kx-ωt) = -iωΨ = -i(E/ħ)Ψ
iħd/dt Ψ = EΨ

dΨ/dx = ikei(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

d dt Ψ = -
ħ²
2m
dx² Ψ

This is the Schrödinger equation.