Confine It and Energy Becomes Discrete
Confine a particle to a box [0, L] of length L. The wavefunction must vanish at both walls, so not just any wave fits: only standing waves pinned at both ends survive. That means only shapes where an integer number of half-wavelengths fit are allowed, and the energy can take only discrete values. In the first figure we watch the standing wave ψₙ(x) inside the box as we change n.
At both walls of the box ψ must be zero. So the only allowed shapes are standing waves pinned to zero at each end, like ψₙ(x) = √(2/L)·sin(n π x / L). Step n through 1, 2, 3, 4 and you see one more node appear each time. The n-th state has n-1 interior nodes, with its ends nailed to the walls.
Now read the energy each standing wave carries as a ladder. The energy that first shows up here is written Eₙ (an energy level), where the subscript n tells you which rung. It shares the letter E with the electric field E but is a different quantity: a discrete energy level. Because Eₙ ∝ n², the rungs sit at ratios 1, 4, 9, 16, and selecting n highlights that bar. You can see the spacing widen as you climb.
Why only integers? Inside the box an integer number of half-wavelengths must fit. The n-th state holds n half-waves, so the wavelength is fixed at λₙ = 2L/n. Drag the slider to change the number of half-waves, and a shape that hits zero at both ends only forms at whole numbers. The boundary condition quantizes λ, and once λ is fixed the energy is quantized too.
The lowest state, n=1, is called the ground state. It is the simplest single hump with no nodes, yet its energy is not zero but a minimum value E₁. This is the zero-point energy. Slide the box width L smaller, and because E₁ ∝ 1/L², the ground energy shoots up fast as the box narrows. Using ℏ = h/2π, E₁ = π²ℏ²/(2 m L²), so confinement forces an energy that can never be zero.
In the end a single integer n fixes everything at once. Choose n and you set the number of humps, the node count (n-1), and the energy Eₙ ∝ n² together. Step n up and down and watch all three readouts change in lockstep. One integer names each allowed state, and that integer is called the quantum number.