Let’s see where we are going

Παρουσίαση με θέμα: "Let’s see where we are going"— Μεταγράφημα παρουσίασης:

1 Let’s see where we are going

2 Maxwell Equation for Light Waves

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

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

5 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 c ∂t2 Δ 8π2m h2 2 ψ (E-V) ψ = 0 Schroedinger equation for electrons

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

7 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

8 p = h/λ

9 p = h/λ c = ω λ

10 p = h/λ c = ω λ 1/λ = ω/c

11 p = h/λ c = ω λ 1/λ = ω/c p = hω/c

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

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

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

15 p = h/λ c = ω λ 1/λ = ω/c p = hω/c p = h2ω2/c2 E = T + V E = ½mv2 + V E = p2/2m + V

16 p = h/λ c = ω λ 1/λ = ω/c p = hω/c p = h2ω2/c2 E = T + V E = ½mv2 + V E = p2/2m + V 2m(E – V) = p2

17 p = h/λ c = ω λ 1/λ = ω/c p = hω/c p = h2ω2/c2 E = T + V E = ½mv2 + V E = p2/2m + V 2m(E – V) = p2 2m(E – V) = h2/λ2

18 p = h/λ c = ω λ 1/λ = ω/c p = hω/c p = 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

19 2 Ψ = 1 ∂2 Ψ c ∂t2 Δ

20 2 Ψ = 1 ∂2 Ψ c ∂t2 Δ Ψ = ψ e -iωt

21 2 Ψ = 1 ∂2 Ψ c ∂t2 Δ Ψ = ψ e -iωt ∂Ψ ∂t = (-iω)ψ e –iωt

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

23 Ψ = ψ e -iωt = (-iω)ψ e –iωt = -ω2 ψ e –iωt Δ Ψ Δ 4π2ω2 c2 2 Ψ =
2 Ψ = 1 ∂2 Ψ c ∂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

24 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 c ∂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

25 Δ 2 ψ = - ω2 c2

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

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

28 p = h/λ c = ω λ 1/λ = ω/c p = hω/c p = 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

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

30

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

32 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

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

34

35

36

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

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

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

40 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 c ∂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

41 H is the Hamiltonian

42 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 c ∂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