= Solution
The classical minimum is \b[$E_{\rm cl}=0$], attained at $x=p=0$. The quantum oscillator has \b[$E_0=\hbar\omega/2$], its <zero-point energy>; simultaneous zero position and <momentum> uncertainties are forbidden.
Let $\ell=\sqrt{\hbar/(m\omega)}$ and $\xi=x/\ell$. The first three normalized <harmonic oscillator eigenfunctions> are
$$
\psi_0=\frac{e^{-\xi^2/2}}{\pi^{1/4}\sqrt\ell},\qquad\psi_1=\sqrt2\xi\psi_0,\qquad\psi_2=\frac{2\xi^2-1}{\sqrt2}\psi_0.
$$
They have respectively zero, one and two nodes, with even, odd and even <parity>. Their phases may be reversed without changing the state.<Quantum Mechanics by David Tong>[Chapter 2: the harmonic oscillator]{ref}
\Image[/past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1-oscillator-states.png]
{title=The first three harmonic oscillator wavefunctions have increasing node count and alternating parity}
{source=paper-1-oscillator-states.py}
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