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Introduction to String Theory

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Introduction to Superconductors

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Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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test

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The Standard Model

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Gravitational Force and the Planck Scale

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Introduction to the Standard Model

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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge

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Quantum Flavordynamics and Quantum Chromodynamics

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Special Unitary Groups and the Standard Model - Part 1

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Special Unitary Groups and the Standard Model - Part 2

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Special Unitary Groups and the Standard Model - Part 3

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Standard Model Lagrangian

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The Higgs Mechanism

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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Last modified: January 26, 2018


Schrodinger Wave Equation
-------------------------
Assume electron is a travelling wave (free particle) defined by:

ψ = Aexpi(kx-ωt))

Spatial derivative:

∂ψ/∂x = ikψ = (ip/h)ψ 

Multiply both sides by ih to get:

-ih∂ψ/∂x = pψ  

∴ (h²/2m)∂²ψ/∂x² = (p²/2m)ψ 

                = Eψ   ... A.
    
This is the TIME INDEPENDENT form.
                                        
Time derivative:

In addition to its role in determining system energies, the Hamiltonian 
operator also generates the time evolution of the wavefunction. 

∂ψ/∂t = -iωψ = (-iE/h)ψ since E = hω

Multiply both sides by ih to get:

∴ ih∂ψ/∂t = Eψ  ... B.

Equating A. with B. we get:
                                                      
-(h²/2m)∂²ψ/∂x² = ih∂ψ/∂t 

This is the TIME DEPENDENT form.

The time-independent Schrodinger equation is the equation describing 
stationary states (atomic or molecular orbitals).  It is only used when 
the Hamiltonian itself is not dependent on time.  Fortunately, the majority 
of interesting problems in quantum mechanics involve stationary states 
with definite total energy and so the time dependent form is not required.

Equation in the presence of a potential:

Time Independent form:

-(h²/2m)∂²ψ(x)/∂x² + U(x)ψ(x) = Eψ(x)

Time Dependent form:

-(h²/2m)∂²ψ(x,t)/∂x² + U(x)ψ(x,t) = ih∂ψ(x,t)/∂t