Particle in a Box Simulator
Explore quantum confinement. See wave functions and probability densities for a particle confined to a 1D infinite potential well.
Particle in an Infinite Square Well
The particle-in-a-box (infinite square potential well) is the simplest exactly solvable problem in quantum mechanics. An electron (or any particle of mass m) is confined to a region 0 ≤ x ≤ L by infinitely high potential walls. Inside the box the potential is zero; outside it is infinite.
Wave Functions
The allowed wave functions are standing waves that fit exactly inside the box: ψ_n(x) = √(2/L) sin(nπx/L). They satisfy ψ=0 at both walls (boundary conditions). The quantum number n (positive integer) counts the number of antinodes (half-wavelengths) in the box.
Energy Quantization
Only certain energies are allowed: E_n = n²π²ℏ²/(2mL²). The energy is proportional to n², so E_2 = 4E_1, E_3 = 9E_1, etc. Unlike classical mechanics, the particle cannot have zero energy — the ground state (n=1) has zero-point energy E_1 > 0.
Probability Density
The probability of finding the particle between x and x+dx is |ψ_n(x)|² dx = (2/L) sin²(nπx/L) dx. For n=1, the particle is most likely at the centre. For n=2, there are two peaks and a node (zero probability) at the centre. Higher n gives more nodes and a more uniform distribution in the classical limit.
| n | Energy E_n | Nodes (ψ=0 inside box) | Most probable positions |
|---|---|---|---|
| 1 | E₁ | 0 | Centre |
| 2 | 4E₁ | 1 (at L/2) | L/4, 3L/4 |
| 3 | 9E₁ | 2 | L/6, L/2, 5L/6 |
| 4 | 16E₁ | 3 | L/8, 3L/8, 5L/8, 7L/8 |
Particle-in-Box Formulas
Allowed Energies
Wave Function
Probability Density
de Broglie Wavelength
| Symbol | Name | Value |
|---|---|---|
| ℏ | Reduced Planck constant | 1.055 × 10⁻³⁴ J·s |
| m_e | Electron mass | 9.109 × 10⁻³¹ kg |
| L | Box length | m (or nm) |
| n | Quantum number | 1, 2, 3, … |
| E_n | Energy of level n | J (or eV) |
| ψ_n | Wave function | m⁻¹/² |
Frequently Asked Questions
Why can't the particle be at rest (zero energy) inside the box?
The Heisenberg uncertainty principle requires Δx·Δp ≥ ℏ/2. Confining the particle to a box of size L means Δx ≤ L, so Δp ≥ ℏ/(2L) > 0. A nonzero uncertainty in momentum means nonzero kinetic energy. This zero-point energy E₁ = π²ℏ²/(2mL²) is a purely quantum effect with no classical analog.
What does a 'node' in the wave function mean physically?
A node is a point where ψ_n = 0, meaning the probability density |ψ|² = 0. The particle has zero probability of being found at that point. Yet it can appear on either side of a node without 'crossing' through it — one of the many counterintuitive features of quantum mechanics.
How does the energy scale with box size L?
E_n ∝ 1/L². Doubling the box length reduces all energies by a factor of 4. This is why electrons confined to smaller structures (quantum dots, molecules) have larger energy gaps and absorb/emit higher-frequency (bluer) light.
What is the classical limit for large n?
For very large n, the probability density |ψ_n|² develops many closely-spaced peaks. Averaged over many peaks, the probability becomes nearly uniform across the box — matching the classical prediction that a bouncing particle spends equal time everywhere. This is Bohr's correspondence principle.
How does this model relate to quantum dots?
Semiconductor quantum dots confine electrons in all three dimensions, effectively creating a 3D box. The three-dimensional energy levels depend on quantum numbers n_x, n_y, n_z. By controlling the dot size (typically 2–10 nm), manufacturers tune the emission wavelength across the visible spectrum — the basis of quantum dot displays (QLED TVs).