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 assume
is a function of both
and
If we assume thatis a product of a function ofwith a function of
then
Substitution of this expression into (1) gives
Divide throughout by
to obtain
Rearrange to get
and
are 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 constant
Ignoring the arbitrary constant
we can write
and
Sinceis arbitrary we can write
and
to 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 is
and the energy corresponding to oscillations in the– direction is
The total energy will be the sum of these:
