Particle in a box (3D)
About this app
A standing matter wave in a glass trough: drag the walls and watch the energy ladder E_n = n²h²/(8mL²) spread live. 1000 position measurements build up |ψ|², and a classical ball shows the difference. Transitions with photon and wavelength, electron versus proton, a prediction question and exam-style calculations.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Why the energy in a box is quantised
In a one-dimensional box with infinitely high walls an electron cannot get out. The wave function ψ must be zero at both walls. So only standing waves with L = n · λ/2 fit in. With the de Broglie relation p = h/λ and the kinetic energy E = p²/(2m) this gives E_n = n²h²/(8mL²). The energy is discrete: it grows with n² and falls with 1/L². Even the ground state has an energy above zero. For an electron at L = 1 nm it is 0.376 eV.
What the app shows
In a glass trough ψ stands as a curtain of light, the probability density |ψ|² lies below it as a ridge, and the energy ladder stands next to it. When you push the walls together, all levels rise at once. Single position measurements collect as glowing grains and gradually fill the shape of |ψ|². A classical ball can be switched on: it runs back and forth and gives an even distribution. For the electron at n = 2 the middle stays empty, because it has no path. An extra view shows tunnelling at a finite wall and is labelled as going beyond the curriculum.
What you can try
Drag a wall and read the new energies on the ladder. Tap a rung to change n. Start a transition and see the photon wavelength on the spectrum. Compare electron and proton: the wave stays the same, the energies shrink by a factor of 1836. Answer the prediction question of where the electron is never found at n = 2, and work through three exam-style calculations yourself. The app recognises typical mistakes and shows the worked solution.
Frequently asked questions
- How do you calculate the energy of a particle in a box?
- In a one-dimensional box with infinitely high walls, E_n = n²h²/(8mL²). Here n is the quantum number, h the Planck constant, m the mass and L the width. For an electron at L = 1 nm this gives E₁ ≈ 6.02 · 10⁻²⁰ J ≈ 0.376 eV.
- Why is the energy in a box never zero?
- A wave with n = 0 would be zero everywhere, so there would be no particle in the box. The smallest allowed wave is half a wavelength with n = 1. It has momentum and therefore the ground-state energy E₁ > 0.
- Does the electron run back and forth in the box?
- No. The electron has no path in the box. |ψ|² says how likely a measurement result is at a given place. At n = 2, |ψ|² is zero in the middle but large to the left and right of it. A ball would have to keep crossing the middle.
- How does the app help with exam preparation?
- It links the formula to the picture: you pull the walls and see the levels rise. It adds a prediction question and three calculations with feedback on typical mistakes and full worked solutions. An explain mode guides you through the scene.
Subject: School physics
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