Quantum wavefunction (Question 2 (10 Marks)) — UPSC Physics Optional CSE 2018 Paper II

The general wave function of harmonic oscillator (one-dimensional) are of the form u_n(x)=\sum_{k=0}^{\infty}a_k y^k e^{-y^2/2} With y=\sqrt{\frac{m\omega}{\hbar}}x, and coefficients a_k are determined by recurrence relations a_{k+2}=\frac{2(k-n)}{(k+1)(k+2)}a_k Corresponding energy levels are E_n=\left(n+\frac{1}{2}\right)\hbar\omega. Discuss the parity of these wave functions. What happens, if the potential for x\leq 0 is infinite (half harmonic oscillator)?