What a learner can do afterwards
- Applies the boundary conditions at the walls and shows why they select integer quantum numbers
- Writes the energy expression and predicts how the spacing changes when the box is made longer or the particle heavier
- Counts nodes for a given level and links node count to energy
- Estimates an absorption wavelength for a conjugated chain treated as a box, and says why the estimate is rough
1 · Read
Picture an electron stuck between two walls it cannot cross, like the electrons that give a dye its colour. Its wave must fall to zero exactly at each wall, like a string pinned at both ends. Only the allowed waves from the last stop survive here, the ones that fit a whole number of half waves between the walls. Each survivor gets a label n = 1, 2, 3, and up. Zero is not allowed: an n = 0 wave would be flat zero everywhere, with no chance of finding the electron anywhere.
Each allowed wave has its own energy, and the energies grow with n squared. Level 2 sits at 4 times the ground energy, and level 3 at 9 times. A longer box or a heavier particle squeezes all the gaps smaller. Notice the gaps grow as you climb, which is the fingerprint of this trap.
Count the nodes to read a level at a glance. Level n has n minus 1 interior nodes, and more nodes always mean higher energy. A dye chain works the same way: take the chain length as the box, and the jump between the highest filled level and the next empty one as the absorbed light. A longer chain gives a smaller jump and hence longer wavelength light, but uneven spacing and repulsion keep the estimate rough.
When you meet a new trap, look at the spacing pattern first. Evenly spaced levels mean a spring-like trap, while widening gaps mean box-like walls. Spectra are fingerprints of the forces inside, so let the gaps tell you the trap shape.
Walls pin the wave to zero, whole half waves pick the levels, and the gaps grow with n.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.