Question

# You do not believe that the ladder operator approach toward solving the Schrodinger equation for the...

You do not believe that the ladder operator approach toward solving the Schrodinger equation for the Harmonic Oscillator potential V(x)=1/2mw^2x^2 gives you a reasonable answer. To verify the calculation, you can use the WKB approximation to determine the ground state energy. Do the results differ and if yes by how much? hint (integral a 0 sqrt(a^2-x^2)dx = pi*a^2/4.

The ladder operator approach gives the ground state energy as

The WKB approximation:

First find the classical turning points(where energy equals potential energy)

This gives the turning points as

and

note that

Then according to WKV approximation:

Therefore

since x2 = -x1

Since this is an even function and it is integrated from limit -x2 to x2

we can writw

or

where

Therefore using the given integral relation, we can write

or the ground state energy is

which matches with the ladder operator approach