Schrodinger Wave Equation

In the light of de Broglie concept of dual character of dual character of matter and the Heisenberg’s uncertainty principle, Bohr’s concept of definite trajectories (orbits) failed. Thus like any other wave, it would be possible to express the motion of the electron in terms of an equation, called the wave equation. The derivation of this equation was carried out by Schrodinger in 1926, making use of the
de Broglie relationshipwhich, therefore, may be called the essential postulate of Schrodinger wave equation. The derivation may be carried out as follows:

Consider the simplest type of wave motion like that of the vibration of a stretched string traveling along the x-axis with a velocity u


                 Vibration of a Stretched string


If  is the amplitude of the wave at any point whose co-ordinate is x, at any time t, then the equation of such a wave motion is

                                    

This differential equation indicates that the amplitude     of the wave at any time traveling with a particular velocity depends upon the displacement x and the time t. In other words, is function of x and t. Hence we may write
                                         

Where  is a function of the co-ordinate x only and is a function of the time t only.

But for the stationary waves.*as occur in the stretched string, we know that

                                            
Where v is the frequency of vibration and A is a constant, equal to the maximum amplitude of the wave.

Subsisting the value of (from eqn. in eqn. we get
                                          
                                          
Differentiating this equation twice w.r.t, we get
                                        
Differentiating eqn. twice w.r.t. x, we get
                                       
Substituting the values of  and form equations and into the equation we get

Or                                             
                                            
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