Schrödinger's equation
H Ψ = i δ Ψ
———
 δ t

The Hamiltonian or Total Energy
KE Kinetic Energy + PE Potential Energy
H =  
-
 
2
——
2 m
 
Δ +
 
V(x,y,z,t)
where:

Δ = 2 = δ2
——
δ x2
+ δ2
——
δ y2
+ δ2
——
δ z2

Schrödinger's equation
 
-
 
2
——
2 m
Δ Ψ(x,y,z, t )  
+
 
 
V(x,y,z,t) Ψ(x,y,z,t)
 
 
= i
 
δ Ψ(x,y,z,t)
————
  δ t

Schrödinger's equation
- in one dimension Ψ(x,t) -
 
-
 
2
——
2 m
δ2 Ψ(x,t)
———
 δ x2
 
+
 
 
V(x,t) Ψ(x,t)
 
 
= i
 
δ Ψ(x,t)
———
 δ t
the de Broglie relation :  λ = h / p the Einstein hypothesis :  ν = E / h
E = p2 / 2 m + V (Kinetic energy + potential energy)
Ψ(x,t) = c1 Ψ1(c,t) + c2 Ψ2(c,t)


Planck’s constant : h = 6.626 068 76 x 10 - 34 J s
= h / 2 π

The probability density that a wave-particle will be found is:
Max Born Postulate
P(x,t) = Ψ*(x,t) Ψ(x,t)

In E = p2 / 2 m + V
if you do: p = - i δ / δx
and E = i δ / δt
you have Schrödinger's equation.