A spherical quantum dot of radius R confines its electron the way a particle-in-a-box confines a particle in 1D, just in 3D spherical geometry. The lowest-energy (ground-state) solution of the infinite spherical well has a radial wavefunction shaped by the spherical Bessel function j₀, and its confinement energy above the bulk conduction band edge is:
ψ(r) ∝ sin(πr/R) / r (ground state, ℓ=0)
ΔE(R) = ℏ²π² / (2·μ·R²) (μ = reduced electron-hole mass)
E_gap(R) = E_bulk + ΔE(R)
λ_emit = 1240 / E_gap(eV) [nm]
- Dot radius R — the physical size of the nanocrystal. The point cloud renders |ψ(r)|² by sampling points with a radial density proportional to the actual squared wavefunction, so shrinking R visibly squeezes the cloud into a tighter, brighter core while the confinement energy (and bandgap) rises as 1/R² — this is the same real relation, just rendered volumetrically instead of as a 1D ladder.
- Material — sets the bulk bandgap and the reduced effective mass μ, which sets how strongly the confinement energy responds to size.
- Probability cloud — toggles the |ψ(r)|² point-cloud visualization versus the bare confining sphere, so you can compare the electron's real spatial spread to the dot's physical boundary.
Real-world relevance: this 3D squeeze-the-electron picture is literally what happens inside a colloidal CdSe or InP nanocrystal during synthesis — controlling growth time controls R, which controls the emitted color, which is how QLED display manufacturers dial in pure red, green and blue from the same base chemistry.