[UPSC IFoS 2016] Schrodinger’s Wave equation and applications (Q173, 30M)

Consider a one-dimensional harmonic oscillator potential V(x) = \frac{1}{2} m \omega_0^2 x^2 where m represents the mass of a particle, \omega_0 is the classical frequency of the oscillator and x is the displacement from equilibrium position. Obtain the solution of the corresponding Schrödinger equation. Using the normalization condition on the solution, show that the ground-state wave function has the form \psi_0(x) = \left( \frac{\alpha}{\pi} \right)^{1/2} e^{-\frac{1}{2} \alpha^2 x^2} \alpha = \left( \frac{m \omega_0}{\hbar} \right)^{1/2}