We set up the Schödinger equation:
The harmonic oscillator is subjected to a potential of the type: \(V(x)=\frac{1}{2}kx^2\)
Solving for the derivative of higher degree, equation (2) can be written:
We simplify equation (3) by defining the following magnitudes:
Substituting (4) and (5) into (3):
Solving equation (6) requires the following change of variable:
Applying the chain rule:
Solving for x from (7) and differentiating with respect to \(\xi\) we obtain:
Substituting (11) and (12) in (9) and solving for \(\frac{d^2\Psi}{dx^2}\):
Substituting (10) and (13) in (6):
Dividing (14) by \(\beta\):
By taking a common factor from the wave function, the Hermite-Gauss equation is obtained, known in Mathematics even before the birth of quantum mechanics.
