p = h/ λ

1. p = h/λ

2. http://people.rit.edu/andpph/photofile-c/splash1728.jpg

3. All we need are Einstein’s two equations E = hv for a photon and E = mc2

4. 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

5. 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

7. image at: www.pnas.org/.../102/42/14952/F2.expansion.html

9. image at: www.pnas.org/.../102/42/14952/F2.expansion.html

15. www.quantum.univie.ac.at/.../c60/c60beug.gif

16. www.macdiarmid.ac.nz/.../images/diffraction.jpg

19. 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

20. ψ 2 ψ = - Δ 1 λ2 ψ 2 ψ = - Δ 2m(E – V)/h2

21. ψ 2 ψ = - Δ 1 λ2 ψ 2 ψ = - Δ 2m(E – V)/h2 ώ = 2πω

22. 2 Ψ = 1∂2 Ψ c2 ∂t2 Δ 2 ψ = - Δ ω2 c2 Ψ = ψ e -iωt 1/λ = ω/c So 1/λ2 = ω2/c2 ∂Ψ ∂t = (-iω)ψ e –iωt ∂2 Ψ ∂t2 = -ω2ψ e –iωt ψ 2 ψ = - Δ Ψ 1 λ2 2 Ψ = - Δ ω2 c2 ψ 2 ψ = - Δ 1 λ2 ψ 2 ψ = - Δ ω2 c2