Physics Harmonic oscillators

Question

I have been trying to use the Schrodinger equation to solve a harmonic oscillator problem, to find expressions for E and B. Just can't finish the end of it.

I have the equation, which is correct

x^2(mw^2−4ℏ^2B^2)−2ℏ^2B−2Em=0
The advice I have been given is this is of the form Ax2+b=0 in which order to be true requires both A and B to be zero. That is two conditions in the two unknowns E and B which allow you to solve the equation.

Really not sure what to do. Up to this line of working so far is correct I have been told.

Answer 1

If the equation is correct and if the stated conditions are correct, then 

 

(mω² - 4h_bar²B²)x² - 2(h_bar²B + Em) = 0

 

Ax² + b = 0

 

(1)   A = mω² - 4h_bar²B² = 0

(2)   b = -2(h_bar²B + Em) = 0

 

From (1) get

 

B = [ω/(2h_bar)] √m

 

From (2) get

 

E = -h_bar²B/m = -(h_bar²/m)[ω/(2h_bar)]√m = -h_barω/(2√m)