1000 Solved Problems in Modern Physics

(Grace) #1

Chapter 3


Quantum Mechanics – II


3.1 Basic Concepts and Formulae ................................


Schrodinger’s equation


i

∂ψ
∂t

=−

^2

2 μ

∇^2 ψ+V(r)ψ (3.1)


^2

2 μ

∇^2 ψ+(V−E)ψ=0 (Time independent equation) (3.2)

Probability densityρ=ψ∗ψ (3.3)

Continuity equation

∂ρ
∂t

+∇.j= 0 (3.4)

wherejis the probability current density.


Normalizationof the wave function



all space

ρdτ=


all space

ψn∗ψndτ= 1 (3.5)

I=


ψmψndτis called the overlapping integral.

Orthogonality


∫+∞

−∞

ψn∗(x)ψm(x)dx= 0 ,ifm =n (3.6)

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