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The Time Independent Schrodinger Equation

The Schrodinger Equation is the fundamental equation of quantum physics and is expressed as follows:
where
  • t represents time,
  • r represents displacement,
  • m is the mass of the particle,
  • i is the square root of minus one and
  • h-bar is Planck's Constant.
is a function of r and t and cannot, in general, be expressed as a function only of 't' multiplied by another function only of 'r'.
However there is a special class of solutions where this condition applies. These represent an important special case. It turns out that we will be able to express general solutions to the Schrodinger Equation as the sum of a series of such special, or 'separable' solutions.

We will now demonstrate how to formulate the two components of the separable solution to Schrodinger's Equation. Substituting into Schrodingers equation gives.

Dividing both sides by gives us:

Notice that we have chosen to force V to be a function only of r, and not of t, in order to create the condition such that we can claim that the first expression in this equation chain is a function only of t, and the second is a function only of r.

We must now conclude that both expressions are equal to a third expression which can be neither a function of r nor of t. In other words they both must be equal to a constant which we have defined here as 'E'.

Reorganizing the above gives us the time dependent factor in the separable solution to the Schrodinger Equation:

and we also have generated the time independent Schrodinger Equation which we were seeking at the outset.

Finding solutions to the time independent Schrodinger Equation will generate the components we need for building time dependent solutions. However this equation has important things to tell us about the behaviour of materials all by itself.

        
 
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