Chapter 3
Quantum Mechanics – II
3.1 Basic Concepts and Formulae
Schrodinger’s equation
i
∂ψ
∂t
= −
2
2μ
∇
2
ψ + V (r)ψ
(3.1)
−
2
2μ
∇
2
ψ + (V − E) ψ = 0 (Time independent equation)
(3.2)
Probability density ρ = ψ
∗
ψ
(3.3)
Continuity equation
∂ρ
∂t
+ ∇. j = 0
(3.4)
where j is the probability current density.
Normalization of the wave function
all space
ρdτ =
all space
ψ
∗
n ψ n dτ = 1
(3.5)
I =
ψ m ψ n dτ is called the overlapping integral.
Orthogonality
+∞
−∞
ψ
∗
n (x)ψ m (x)dx = 0, if m = n
(3.6)
131
Quantum Mechanics – II
3.1 Basic Concepts and Formulae
Schrodinger’s equation
i
∂ψ
∂t
= −
2
2μ
∇
2
ψ + V (r)ψ
(3.1)
−
2
2μ
∇
2
ψ + (V − E) ψ = 0 (Time independent equation)
(3.2)
Probability density ρ = ψ
∗
ψ
(3.3)
Continuity equation
∂ρ
∂t
+ ∇. j = 0
(3.4)
where j is the probability current density.
Normalization of the wave function
all space
ρdτ =
all space
ψ
∗
n ψ n dτ = 1
(3.5)
I =
ψ m ψ n dτ is called the overlapping integral.
Orthogonality
+∞
−∞
ψ
∗
n (x)ψ m (x)dx = 0, if m = n
(3.6)
131