The Separation of Variables Method - The Simple Harmonic Oscillator

In two dimensions Schrodinger's equation takes the form(1)

Because x and y appear in the equation, we must assumeis a function of bothandIf we assume thatis a product of a function ofwith a function ofthenSubstitution of this expression into (1) gives

Divide throughout byto obtain

Rearrange to get

andare independent variables, independent of each other, The left hand side is a function ofonly and the right hand side is a function ofonly, so both sides must be equal to a constantIgnoring the arbitrary constantwe can write

and

Sinceis arbitrary we can writeandto obtain

and

These two equations can be solved separately as for the one dimensional simple harmonic oscillator and the two solutions multiplied to give the general solution,to (1) . The energy of oscillations in the– direction isand the energy corresponding to oscillations in the– direction isThe total energy will be the sum of these: