Question

Photons are the force carriers and source of electromagnetic interactions between electrons and protons. How are electromagnetic interactions described by wave mechanics? (i.e. Schrodinger's equation, the Hamiltonian operator, the Born probability density function, etc).

Answer 1

Time independent Schrödinger wave equation

Consider a system of stationary waves to be associated with the particle. Let be the wave displacement for the de-Broglie waves at any location at time t. Then according to Maxwell’s wave equation, the differential equation of the wave motion in three dimension can be written as      

Where and u is the velocity of the wave

We have ψ as a periodic displacement in terms of time

i.e.

Where ψ_o,, is the amplitude of the wave. It is function of r not of time t.

The above equation is called Schrödinger time independent wave equation. The quantity ψ is wave function. where

Time dependent Schrödinger wave equation

Time dependent Schrödinger wave equation can be obtained by eliminating Energy operator from equation above

Energy operator

or

The above equation is called Schrödinger time dependent wave equation, Also

The wave equation describes the dynamics of quantum mechanical systems via the wave function. The t The Schrodinger equation is the name of the basic non-relativistic wave equation used in one version of quantum mechanics to describe the behaviour of a particle in a field of force. There is the time dependant equation used for describing progressive waves, applicable to the motion of free particles. And the time independent form of this equation used for describing standing waves. Trajectory, the positioning, and the energy of these systems can be retrieved by solving the Schrödinger equation.

On a very simple level, we can think of electrons as standing matter waves that have certain allowed energies. Schrödinger formulated a model of the atom that assumed the electrons could be treated at matter waves. While we won't be going through the math in this article, the basic form of Schrödinger's wave equation is as follows:

H^ψ=Eψ

ψ is called a wave function; H^ with, hat, on top is known as the Hamiltonian operator; and E is the binding energy of the electron.

One very important use of the wavefunction comes from the Born interpretation, which derives information about the location of a particle from its wavefunction.

Quite simply, it states that the square modulus of the wavefunction, |Ψ|² , at any given point is proportional to the probability of finding the particle at that point. (The quantity |ψ|² is thus a probability density.)

This is simply illustrated in a 1-dimensional situation, when we can state that if the wavefunction of a particle has a value ψ at the point x, then the probability of finding the particle between x and x + dx is proportional to |ψ|² dx . Thus the probability of finding the particle between two points a and b is proportional to the integral of the square modulus of the wavefunction, evaluated between limits of a and b: