particle in a box explained
Shut a particle inside the box [0, L]. The wave function has to be 0 at the wall, so an arbitrary wave cannot enter and only a standing wave tied at both ends remains. Only shapes whose half-waves fit a whole number of times are allowed, and the energy can take only separated values. The first picture changes ψₙ(x) inside the box.
At both walls ψ must be 0. The allowed shape is only ψₙ(x) = √(2/L)·sin(n π x / L), a standing wave whose ends are 0. Raise n through 1, 2, 3, 4 and the nodes (node) grow one by one. The nth state holds n-1 nodes inside, and the ends stay tied to the walls.
Look at the energy of each standing wave as a ladder. The energy here is Eₙ, an energy level, and the subscript says which rung it is. It is a different quantity from the electric field E, a level that comes in separated steps. Because Eₙ ∝ n², the rungs spread in the ratio 1, 4, 9, 16, and choosing n highlights that rung. Higher up, the gaps grow wider.
Why only whole numbers. Inside the box a whole number of half-waves has to fit. The nth state holds n half-waves, so the wavelength is fixed as λₙ = 2L/n. Change the count of half-waves and a shape with both ends at 0 appears only for a whole number. The boundary condition quantizes the wavelength, and once the wavelength is fixed the energy is quantized too.
The lowest n=1 is called the ground state. It is a simple hump with no node, yet the energy is not 0 and it carries a minimum E₁. That is the zero-point energy. Shrink the box width L and E₁ ∝ 1/L², so a narrower box pushes the floor up fast. Using ℏ = h/2π gives E₁ = π²ℏ²/(2 m L²), and confinement refuses to let the energy become 0.
One whole number n sets everything at once. Set n and the number of humps, the number of nodes (n-1), and the energy Eₙ ∝ n² are fixed together. Raise and lower n and the three values move together. One whole number names each allowed state, so it is called a quantum number.