Difference between Particle in a Box and Harmonic Oscillator approximations
Quantum numbers pop up when you solve Schrödingers wave equation for systems whose solutions are discrete due to the presence of a restraining potential. There is some degree of arbitrariness to what the range of quantum numbers might be, because it depends on how you define the quantum number. 

You could just as well define the quantum number and associated energy levels for a 1D  pib as $$\begin{align} E_{n'}&amp;=k(n'+1)^2 \\&amp;=k n'^2 +2kn' + k\end{align} $$ where $$k=\frac{\pi^2 \hbar^2}{2mL^2}$$ Here $n'$ defines a "new" quantum number related to the original one as $n'=n-1$ and therefore ranges from 0 to $\infty$.

For the harmonic oscillator, you could similarly rewrite the energy as 

$$E_n = \hbar \omega n'$$

where $n'$ ranges in integer increments from $\frac12$ to $\infty$.

Physically speaking, the non-zero ground-state energy of both the pib and harmonic oscillators is usually  attributed to the Heisenberg Uncertainty Principle, because a particle with no energy would have no velocity and therefore would be completely unrestrained in position. There are of course plane waves, but this would be a strange plane wave, with no amplitude.