Sunday, January 11, 2009

What is Schrodinger's Equation ?

In quantum mechanics, the Schrodinger equation plays the same role as the Newton's second law does in classical mechanics. The Schrodinger equation was such a breakthrough made in the history of mankind which revolutionized the world of physics. The total energy of a particle is obtained by adding its kinetic energy to its potential energy. This is called the Hamiltonian of the particle. A simple way to get the most respected equation which was written by Erwin Schrodinger in the first quarter of the 20th century is to replace the energies by operators. The total energy is replaced by ( i hbar del/ del t ), the kinetic energy is replaced by ( hbar^2/(2m) * laplacian^2) and the potential energy is replaced by the potential energy operator, V. Then, we get the time dependent schrodinger equation, which can be operated to a wave function psi as :
( i hbar del/ del t ) psi = (- hbar^2/(2m) * laplacian^2) psi + V psi
This equation is the backbone of quantum mechanics. If we consider a free particle, the potential energy of the particle is zero. Then this equation explains how a free particle evolves with time. The solution to this equation under a suitable potential always gives the time evolution of a quantum system under that potential. If we replace ( i hbar del/ del t ), by the energy operator E, the above equation takes a form:
( E ) psi = (- hbar^2/(2m) * laplacian^2) psi + V psi
This is time-independent (stationary) Scrodinger equation. The solution to this equation gives the stationary state of a system depending upon the nature of the potential. For example, if we take the potential to be parabolic, V = (1/2)m (omega)^2 (x)^2, in the time-independent schrodinger equation, we will get the stationary states of this potential, which is popularly known as the stationary states of a harmonic oscillator.

1 comment:

Dale ritter said...

Thanks for the notes on particle physics in terms of the harmonic oscillator model with the Schrodinger wavefunction. This mathematical analysis now has a new differential expansion of a psi's energy to 29 orders, the GT integral atomic model.
Disclosed in U.S. District (NM) Court of 04/02/01, the motion titled The Solution to the Equation of Schrodinger displayed how the GT atomic topofunc generates the exact, picoyoctometric topological data image of an atom, psi. The h-bar and more pymtechnical particles are viewable online at http://www.symmecon.com .
CRQT functions work by integration of relativistic formulas into the quantum equations for waves, giving models of waveparticles pulsating at [ e = h (nhu) ] frequencies by relativistic gains and losses of four types of force particles: chronons, probablons, varietons (magnetic), gravitalons.
Thanks for pointing out the waveparticle fundamentals, and I hope you enjoy the solution to the time-dependent Schrodinger equation as well as the complete CRQT grand unified theory of matter and energy in the book, The Crystalon Door.