This  is a very common operator used in quantum mechanics. It is represented by(pronounced as ‘del squared’) and is defined as

                                          
                                
This Schrodinger wave equation viz.,
                               

May be written in terms of Laplacian operator as
                                   
Hamiltonian operator. Schrodinger wave equation as written above may be rewritten in the form
                                          

or                                         

or                

or                

This equation implies that the operation carried on the function , is equal  to the total energy multiplied with the function.
Theoperator is called Hamiltonian operator and is represented by . Thus

                                                                    
Eqn may, therefore, be written as
                                                                              
This is another short-hand form of writing the Schrödinger wave equation.
As mentioned earlier, is called the eigen function and E called eigen value.

Such an equation is, therefore, called eigen value equation. Thus for Schrödinger wave equation, we can write
          (Energy operator)(Wave function) = (Energy)X(Wave function)
This result can be generalized as it is equally applicable to all other observables i.e. measurable properties of a system. Thus, in general, we can write

(Operator corresponding to an observable) (Wave function)
                                                              = (Value of observable)X( Wave function)

Thus if we know both the wave function and the operator for the observable and if the wave function is an eigen function, we can predict the mechanics given in the next section.


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