The Oscillator according to Wave Mechanics

In this section we shall obtain the solution of the wave equation for the linear harmonic oscillator. The equation is 

Here A is the proper-value parameter and we have to find the values of A for which the equation has a finite and unique solution throughout all space. 

We can write down one solution of the equation at once, namely, the so-called Gaussian error function 

In order to find tlu . riunaining solutions of the wave equation, it is convenient to assunu that ijj is of the form 

If we substitute tbe above expression in the differential equation, a brief calculation gives the following differential equation for v  

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