A particle on a spring is pulled back harder the further it goes from the balance point. The pull is proportional to the distance, so the potential energy is the parabola U = \dfrac{1}{2}kx^2.
The walls of the potential well stood straight up. These walls rise smoothly instead. They grow higher the further out one goes, so how far the particle can reach is set by its energy.
Classically, fixing E fixes the swing, and the particle runs back and forth inside it. E can be made as small as one likes, and at 0 the particle simply sits still at the balance point. Nothing is discrete so far.
The Schrödinger equation is solved for this parabola. As the previous article showed, what decides which way the curve bends is whether E is above or below U.
Inside, where E > U, the wave function oscillates. Outside, where E < U, it does not oscillate and falls away instead. The boundary is where E meets the parabola, which is the turning point in the classical picture.
For most values of E the outside part does not fall away but diverges, which means nothing far from the centre. Keeping only the solutions that stay finite narrows E down to a discrete set. In the well the condition was a node at each end; here it is that the function must go to 0 far away.
Lining up the values of E that remain gives E_n = \left(n + \dfrac{1}{2}\right)h\nu, counting n from 0, where \nu = \dfrac{1}{2\pi}\sqrt{k/m} is the classical frequency of the spring itself.
In the potential well E_n went as n^2, so the gaps opened out toward the top. Here they do not. The gap between neighbouring levels is h\nu everywhere, so the levels are evenly spaced.
The difference comes from the shape of the walls. The well keeps the same width, so a higher level had to pack a shorter wavelength into it. The parabola widens as it rises, and the packing and the widening balance, which leaves every gap the same.
At n = 0 there is still E_0 = \dfrac{1}{2}h\nu left. Even the lowest state does not come down to 0. This is called the zero-point energy.
The reason it cannot sit still is the one met in the well. Resting exactly at the balance point would mean the position and the momentum are both fixed at once, and the uncertainty principle does not allow it.
The wave function of the lowest state is a single hump centred on the balance point. It has no node, and it seeps a little past the classical turning points on both sides.
Blackbody radiation started from limiting the energy of a vibration to whole multiples of h\nu, which made the curve match the measurements. Planck put that in by hand, as an assumption.
What has just been solved is that very vibration. The levels stand h\nu apart, and what can be given or taken comes in units of h\nu. The assumption has come out of the equation.
The only difference is the \dfrac{1}{2}h\nu. Planck wrote E = nh\nu, and the solution is E_n = \left(n + \dfrac{1}{2}\right)h\nu. That \dfrac{1}{2}h\nu is added to every level by the same amount, so it never appears in what is exchanged, and the blackbody curve is unchanged.