Let’s see where we are going

Maxwell Equation for Light Waves

Maxwell Equation for Light Waves Δ 2 Ψ = 1 ∂2Ψ c2 ∂t2

Maxwell Equation for Light Waves Δ 2 Ψ = 1 ∂2Ψ c2 ∂t2 Δ 8π2m h2 2 ψ + (E-V) ψ = 0

Δ 2 Ψ = 1 ∂2Ψ c2 ∂t2 Δ 8π2m h2 2 ψ + (E-V) ψ = 0 Maxwell Equation for Light Waves Δ 2 Ψ = 1 ∂2Ψ c2 ∂t2 Δ 8π2m h2 2 ψ + (E-V) ψ = 0 Schroedinger equation for electrons

Δ 8π2m h2 2 ψ + (E-V) ψ = 0 -h2 2m Δ 2 + V ψ = E ψ h = h/2π

Δ 8π2m h2 2 ψ + (E-V) ψ = 0 -h2 2m Δ 2 + V ψ = E ψ H ψ = E ψ h = h/2π H ψ = E ψ H is the Hamiltonian

p = h/λ

p = h/λ c = ω λ

p = h/λ c = ω λ 1/λ = ω/c

p = h/λ c = ω λ 1/λ = ω/c p = hω/c

p = h/λ c = ω λ 1/λ = ω/c p = hω/c p2 = h2ω2/c2

p = h/λ c = ω λ 1/λ = ω/c p = hω/c p2 = h2ω2/c2 E = T + V

p = h/λ c = ω λ 1/λ = ω/c p = hω/c p2 = h2ω2/c2 E = T + V E = ½mv2 + V

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

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

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

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

2 Ψ = 1 ∂2 Ψ c2 ∂t2 Δ

2 Ψ = 1 ∂2 Ψ c2 ∂t2 Δ Ψ = ψ e -iωt

2 Ψ = 1 ∂2 Ψ c2 ∂t2 Δ Ψ = ψ e -iωt ∂Ψ ∂t = (-iω)ψ e –iωt

Ψ = ψ e -iωt = (-iω)ψ e –iωt = -ω2 ψ e –iωt Δ 2 Ψ = 1 ∂2 Ψ c2 ∂t2 ∂Ψ 2 Ψ = 1 ∂2 Ψ c2 ∂t2 Δ Ψ = ψ e -iωt ∂Ψ ∂t = (-iω)ψ e –iωt ∂2 Ψ ∂t2 = -ω2 ψ e –iωt

Ψ = ψ e -iωt = (-iω)ψ e –iωt = -ω2 ψ e –iωt Δ Ψ Δ 4π2ω2 c2 2 Ψ = 2 Ψ = 1 ∂2 Ψ c2 ∂t2 Δ Ψ = ψ e -iωt ∂Ψ ∂t = (-iω)ψ e –iωt ∂2 Ψ ∂t2 = -ω2 ψ e –iωt ω = 2πω Ψ Δ 2 Ψ = - 4π2ω2 c2

Ψ = ψ e -iωt = (-iω)ψ e –iωt = -ω2 ψ e –iωt ψ Δ Ψ Δ 4π2ω2 c2 Δ 4π2ω2 2 Ψ = 1 ∂2 Ψ c2 ∂t2 Δ Ψ = ψ e -iωt ∂Ψ ∂t = (-iω)ψ e –iωt ∂2 Ψ ∂t2 = -ω2 ψ e –iωt ω = 2πω Ψ Δ 2 Ψ = - 4π2ω2 c2 ψ Δ 2 ψ = - 4π2ω2 c2

Δ 2 ψ = - ω2 c2

Δ 2 ψ = - ω2 c2 1/λ = ω/c So 1/λ2 = ω2/c2

Δ 2 ψ = - ω2 c2 1/λ = ω/c So 1/λ2 = ω2/c2 ψ Δ 2 ψ = - 1 λ2

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

Maxwell Equation for Light Waves Δ 2 Ψ = 1 ∂2Ψ c2 ∂t2 Δ 8π2m h2 2 ψ + (E-V) ψ = 0

E = T + V E = ½mv2 + V E = p2/2m + V p → -ih(m)-1∂/∂x x→ x E = - h2 (2m)-1∂2/∂x2 + V E = - h2 (8π2m)-1∂2/∂x2 + V

E = - h2 (8π2m)-1∂2/∂x2 + V Δ 8π2m h2 2 ψ + (E-V) ψ = 0

ψ ψ Δ ω2 c2 Δ 1 Δ 1 2 ψ = - 1/λ = ω/c So 1/λ2 = ω2/c2 2 ψ = - λ2

ψ ψ Δ ω2 c2 Δ 1 Δ 1 2 ψ = - 1/λ = ω/c So 1/λ2 = ω2/c2 2 ψ = - λ2

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

H is the Hamiltonian

Ψ = ψ e -iωt = (-iω)ψ e –iωt = -ω2 ψ e –iωt ψ Δ Ψ Δ ω2 c2 Δ ω2 c2 2 Ψ = 1 ∂2 Ψ c2 ∂t2 Δ Ψ = ψ e -iωt ∂Ψ ∂t = (-iω)ψ e –iωt ∂2 Ψ ∂t2 = -ω2 ψ e –iωt Ψ Δ 2 Ψ = - ω2 c2 ψ Δ 2 ψ = - ω2 c2