p = h/λ
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Einstein’s two equations All we need are Einstein’s two equations E = hv for a photon and E = mc2
vλ = c (vel = freq x wavelength) v/c = 1/λ Einstein’s Relations E = hv E = mc2 hv = mc2 hv = (mc)c hv = pc (p = mv momentum) h(v/c) = p vλ = c (vel = freq x wavelength) v/c = 1/λ h/λ = p de Broglie’s relation
de Broglie’s relation for the wavelength of a particle – originally the electron p = h/λ But it applies to any particle even a much heavier particle such as C60
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image at: www.pnas.org/.../102/42/14952/F2.expansion.html
image at: www.exo.net/.../patternsnature2004.htm
image at: www.pnas.org/.../102/42/14952/F2.expansion.html
www.quantum.univie.ac.at/.../c60/c60beug.gif
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p = h/λ c = ω λ 1/λ = ω/c p = hω/c p2 = h2ω2/c2 E = T + V E = ½mv2 + V E = p2/2m + V 2m(E – V) = p2 2m(E – V) = h2/λ2 2m(E – V)/h2 = 1/λ2
ψ Δ 2 ψ = - 1 λ2 ψ Δ 2 ψ = - 2m(E – V)/h2
ψ Δ 2 ψ = - 1 λ2 ψ Δ 2 ψ = - 2m(E – V)/h2 ώ = 2πω
Ψ = ψ e -iωt = (-iω)ψ e –iωt = -ω2 ψ e –iωt ψ ψ ψ Δ Δ ω2 c2 Δ 1 Ψ Δ ω2 2 Ψ = 1 ∂2 Ψ c2 ∂t2 Δ Δ 2 ψ = - ω2 c2 Ψ = ψ e -iωt 1/λ = ω/c So 1/λ2 = ω2/c2 ∂Ψ ∂t = (-iω)ψ e –iωt ∂2 Ψ ∂t2 = -ω2 ψ e –iωt ψ 2 ψ = - Ψ Δ 1 λ2 Δ 2 Ψ = - ω2 c2 ψ Δ 2 ψ = - 1 λ2 ψ Δ 2 ψ = - ω2 c2