1. The Schrödinger Recipe

2. All you need to remember is the recipe to obtain the Quantum Mechanical Hamiltonian is dead simple – express the total energy in momenta and coordinates
p2/2m + V = E

3. and then replace the momenta by
All you need to remember is the recipe to obtain the Quantum Mechanical Hamiltonian is dead simple – express the total energy in momenta and coordinates
p2/2m + V = E
and then replace the momenta by
pq → -iħ ∂/∂q

4. ħ is Dirac’s constant h/2π
All you need to remember is the recipe to obtain the Quantum Mechanical Hamiltonian is dead simple – express the total energy in momenta and coordinates
p2/2m + V = E
and then replace the momenta by
pq → -iħ ∂/∂q
q → q
ħ is Dirac’s constant h/2π

6. ħ is Dirac’s constant h/2π
All you need to remember is the recipe to obtain the Quantum Mechanical Hamiltonian is dead simple – express the total energy in momenta and coordinates
p2/2m + V = E
and then replace the momenta by
pq → -iħ ∂/∂q
Where
ħ is Dirac’s constant h/2π
and i = √-1

7. Let’s recap

8. What did Schroedinger do?

9. Maxwell Equation for Light Waves

10. Maxwell Equation for Light WavesΔ
2 Ψ =
1 ∂2Ψ
c ∂t2

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

12. 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

13. Δ
8 π 2m h2
2Ψ (E-V) Ψ = 0

14. Δ
8 π 2m h2
2Ψ (E-V) Ψ = 0
ħ = h/2π Δ 2m ħ2
2Ψ (E-V) Ψ = 0

15. Δ 2m ħ2
2Ψ (E-V) Ψ = 0

16. Δ 2m ħ2
2Ψ (E-V) Ψ = 0 Δ ħ2 2m
2Ψ + (E-V) Ψ = 0

17. 2m Δ ħ2 2Ψ + (E-V) Ψ = 0 Δ ħ2 2Ψ + (E-V) Ψ = 0 2m Δ ħ2 2 + V Ψ = EΨ 2m―
2 + V Ψ = EΨ

18. 2m Δ ħ2 2Ψ + (E-V) Ψ = 0 Δ ħ2 2Ψ + (E-V) Ψ = 0 2m Δ ħ2 2 + V Ψ = EΨ 2m―
2 + V Ψ = EΨ
H Ψ = EΨ

19. H is called the HamiltonianΔ 2m ħ2
2Ψ (E-V) Ψ = 0 Δ ħ2 2m
2Ψ + (E-V) Ψ = 0 Δ ħ2 2m ―
2 + V Ψ = EΨ
H Ψ = EΨ
H is called the Hamiltonian

20. Now for a bit of Magic!

21. Start off with the Total Energy
= Kinetic Energy + Potential Energy
E = T + V

22. T + V = E

23. T + V = E
½mv2 + V = E

24. T + V = E
½mv2 + V = E
p2/2m + V = E

25. T + V = E
½mv2 + V = E
p2/2m + V = E
p2 = px2 + py2 + pz2

26. T + V = E
½mv2 + V = E
p2/2m + V = E
p2 = px2 + py2 + pz2
px → -iħ ∂/∂x

27. T + V = E
½mv2 + V = E
p2/2m + V = E
p2 = px2 + py2 + pz2
px → -iħ ∂/∂x py → -iħ ∂/∂y

28. T + V = E
½mv2 + V = E
p2/2m + V = E
p2 = px2 + py2 + pz2
px → -iħ ∂/∂x py → -iħ ∂/∂y pz → -iħ ∂/∂z

29. Δ T + V = E ½mv2 + V = E p2/2m + V = E p2 = px2 + py2 + pz2
px → -iħ ∂/∂x py → -iħ ∂/∂y pz → -iħ ∂/∂z Δ
2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2

30. Δ Δ T + V = E ½mv2 + V = E p2/2m + V = E p2 = px2 + py2 + pz2
px → -iħ ∂/∂x py → -iħ ∂/∂y pz → -iħ ∂/∂z Δ
2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
p2 = - ħ Δ

31. Δ Δ Δ T + V = E ½mv2 + V = E p2/2m + V = E p2 = px2 + py2 + pz2
px → -iħ ∂/∂x py → -iħ ∂/∂y pz → -iħ ∂/∂z Δ
2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
p2 = - ħ Δ -ħ Δ
+ V = E 2m

32. H is called the HamiltonianΔ 2m ħ2
2Ψ (E-V) Ψ = 0 Δ ħ2 2m
2Ψ + (E-V) Ψ = 0 Δ ħ2 2m ―
2 + V Ψ = EΨ
H Ψ = EΨ
H is called the Hamiltonian

33. All you need to remember is the recipe to obtain the Quantum Mechanical Hamiltonian is dead simple – express the total energy in momenta and coordinates
p2/2m + V = E

34. and then replace the momenta by
All you need to remember is the recipe to obtain the Quantum Mechanical Hamiltonian is dead simple – express the total energy in momenta and coordinates
p2/2m + V = E
and then replace the momenta by
pq → -iħ ∂/∂q

35. ħ is Dirac’s constant h/2π
All you need to remember is the recipe to obtain the Quantum Mechanical Hamiltonian is dead simple – express the total energy in momenta and coordinates
p2/2m + V = E
and then replace the momenta by
pq → -iħ ∂/∂q
Where
ħ is Dirac’s constant h/2π
and i = √-1

37. Δ
8 π 2m h2
2Ψ (E-V) Ψ = 0
ħ = h/2π Δ 2m ħ2
2Ψ (E-V) Ψ = 0 Δ ħ2 2m
2Ψ + (E-V) Ψ = 0

38. 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

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

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

41. Δ 8π2m h2 2 ψ + (E-V) ψ = 0 h2 8π2m Δ 2 ψ + (E-V) ψ = 0 h2 8π2m Δ―
2 ψ + (E-V) ψ = 0
ħ = h/2π