The differential equation for the 1D Harmonic Oscillator is.
Solving the HO Differential Equation *
By working with dimensionless variables and constants, we can see the basic equation
and minimize the clutter.
We use the energy in terms of
We define a dimensionless coordinate.
The equation becomes.
(Its probably easiest to just check the above equation by substituting as below.
Now we want to find the solution for
which has the solution (in the large
This exponential will dominate a polynomial as
so we can write our general solution as
is a polynomial.
Take the differential equation
into it to get
This is our differential equation for the polynomial
as a sum of terms.
Plug it into the differential equation.
We now want ot shift terms in the sum so that we see the coefficient of
To do this, we will shift the term
down two steps in the sum.
It will now show up as
(Note that in doing this shift the first term for
get shifted out of the sum.
This is OK since
is zero for
For the sum to be zero for all
, each coefficient of
must be zero.
and we have a recursion relation giving us our polynomial.
But, lets see what we have. For large
The series for
has the coefficient of
the coefficient of
So our polynomial solution will approach
and our overall solution will not
We must avoid this.
We can avoid the problem if the series terminates and does not go on to infinite
The series will terminate if
for some value of
Then the last term in the series will be of order
The acceptable solutions then satisfy the requirement
Again, we get quantized energies when we satisfy the boundary conditions at infinity.
The ground state wavefunction is particularly simple, having only one term.
by normalizing the wavefunction.