Question 4.P.9: Find the energy levels of a particle of mass m moving in a o...

This is an asymmetric harmonic oscillator potential in which the particle moves only in the region x > 0. The only acceptable solutions are those for which the wave function vanishes at x = 0. These solutions must be those of an ordinary (symmetric) harmonic oscillator that have odd parity, since the wave functions corresponding to the symmetric harmonic oscillator are either even (n even) or odd (n odd), and only the odd solutions vanish at the origin, \psi _{2n+1} (0)=0 (n=0,1,2,3,…). Therefore, the energy levels of this asymmetric potential must be given by those corresponding to the odd n energy levels of the symmetric potential, i.e.,

E_{n} =\left[(2n+1)+\frac{1}{2}\right] \hbar \omega =\left (2n+\frac{3}{2}\right) \hbar \omega         (n=0,1,2,3,…).             (4.284)