Classically, Hooke’s law gives the force as a function of the position,
that substituted in
When calling the solution of the previous equation is where and are integration constants that depend on the initial conditions of the oscillation.
To prove our guess, let us substitute this solution, , on the equation, and Thus, substituting in the Schrödinger equation results in Since this equation must be valid for all , and . The first of these relations is the definition of and the second equation set the value of the energy for this solution,
In order to explore the higher energy solutions, the following variable change is applied . Now, and . Thus, the Schrödinger equation becomes Renaming the Schrödinger equation is reduced to For large , in this case, large , just as in the previous case, the solution should be modulated by a function like so the solution can be written as . Substituting in the previous equation, the differential equation for can be obtained
The previous equation can be solved by using a power series expansion,
Substituting the previous expressions in the differential equation,
For the previous expression to be independent of , each square bracket must be zero; thus, , , and from where the series recurrent relation can be deduced
If no restriction is placed on , it can be proved that the series represents an exponential function of the form, implying that is a diverging solution for large . However, if for a given . The previous series will stop at the given and the function becomes a polynomial. These polynomials are called Hermite polynomials. Therefore, the energy of the oscillator is quantized accordingly with for The corresponding normalized wave functions are written as
Where
Graph of the first few Hermite Polynomials
Corresponding Wave Functions

