[UPSC CSE 2026] Schrodinger’s Wave equation and applications (Q104, 10M)

Obtain the allowed eigenenergies of the half 1-D simple harmonic oscillator defined by the Hamiltonian H = \frac{p^2}{2m} + V(x), \quad \text{where} V(x) = \begin{cases} \dfrac{1}{2}m\omega^2 x^2 & \text{for } x \ge 0 \\ \infty & \text{for } x < 0 \end{cases} Express its eigenstates in terms of eigenstates of the full 1-D oscillator.