Wave Mechanics- Schrodinger Wave Equation

Schrodinger Wave Equation

            Wave mechanics is an elementary formulation of quantum theory that is centered on a wave function Ψ(x,t). Position and momentum of a particle are deduced from Ψ.

   
         Austrian physicist Erwin Schrodinger lays the foundations of quantum wave mechanics. In a series papers he describes his partial differential equation that is the basic equation of quantum mechanics and bears the same relation to the mechanics of the atom as Newton's equations of motion bear to planetary astronomy.

The equation-

 

 The appropriate wave equation is the one-dimensional Schrodinger equation,

  ................(1)

With the generalization to three-dimensions leading to the Laplacian operator in place of 2 x (cf. Maxwell’s equation). Which is time dependent Schrodinger Equation. Particle is moving in a region of varying potential V(x). 

• Time dependent Schrodinger Equation
• Time independent Schrodinger Equation

For the simplification, we will take Ψ(x,t) = ψ(x).T(t) .
Here capital Ψ is used as time dependent Wave function where as lower latter ψ is used as time-independent wave function. 

Now putting the value of Ψ = ψT in the equation (1), we get

Here the left hand side is the function of 't' and the right hand side is the function if 'x' only.
If both sides are equal to a constant, E, say

Where E= ћw.  The complex exponential function can not be replaced by a real sine or cosine. nor is exp(+iωt). 

The spatial part of the Schrodinger Equation becomes 

This is Time independent Schrodinger Equation. The solution of this equation can be given by 

Thus the solution of the time-independent Schrodinger Equation describes the state of particle with a definite energy,known as stationary states.