quantum harmonic oscillator explained
Put a particle in a soft bowl V(x) = ½ m ω² x², not between hard walls. In the box the energy rungs ran 1, 4, 9, 16 and spread as they climbed, but here the gap is the same ΔE = ℏω from the first rung to the last. The levels are Eₙ = (n + ½)ℏω, and the rungs stand in an even row. The first picture sets the bowl V(x) next to that even ladder, and changing the stiffness moves the two together.
Change the stiffness ω of the bowl V(x) = ½ m ω² x². This ω is the angular frequency of the oscillator, the same ω used in classical simple harmonic motion (SHM). Raise ω and the bowl grows narrow and steep, and the ladder on it stretches with it. The rungs always sit at ΔE = ℏω, and that value is shown beside them. Unlike the box values 1, 4, 9, 16, the gaps never bunch up.
See which wave function ψₙ(x) each rung carries. Hₙ is a Hermite polynomial, a different quantity from the Hamiltonian H. ψₙ is a Gaussian hump times Hₙ(x), so the nth state has exactly n nodes (node). Raise n from 0 to 4 and the hump ripples inside the bowl while the nodes grow one by one.
Climb to a high rung and see where the quantum meets the classical. A classical oscillator slows near the turning point and lingers, so the probability of being there is largest as ∝ 1/√(xmax² − x²). Raise n and the rippling peaks of the quantum |ψₙ(x)|² follow that classical curve. When n is small the two differ a lot, and when n is large the probability piles up at both ends and they nearly overlap. That is the correspondence principle.
The same particle, a different bowl, set side by side. The box has hard walls, so Eₙ ∝ n², and the rungs spread like 1, 4, 9, 16 as they climb. The harmonic oscillator is a soft bowl, so Eₙ = (n + ½)ℏω, and the gap stays ΔE = ℏω all the way. Alternate the two ladders and it is plain how the shape of the bowl sets the spacing.
Look closely at the ground n=0. It is a Gaussian hump with no node, yet the energy is not 0 but E₀ = ½ℏω. That is the zero-point energy, so even the calmest state never fully stops. Change ω and the width of the hump changes, but E₀ = ½ℏω stays at half the height of the first gap and does not touch 0. As long as the particle is confined, the floor energy cannot be 0.