Question

Find the solution to the “half” harmonic oscillator:

v ( x ) = 03 x < o

= $kx2 x > 0

Compare the energy values and wave functions with those of the full

harmonic oscillator. Why are some of the full solutions present and some

missing in the “half” problem?

Answer 1

For full harmonic oscillator energy value $E_n = (n + \frac{1}{2})hω$

For half harmonic oscillator energy value $E_n = (2n +1+ \frac{1}{2} ) hω$

ω is angular frequency

$h=\frac{h}{2} \times \pi$

Here n is replaced by (2n+1) because only odd solution are possible in case of half harmonic oscillator because wave function must vanish at origin as there potential is infinite.

That's why odd integers give full solution And others are missing.