Particle in a Box Calculator
Computes the quantised energy levels of a particle confined in a 1D, 2D, or 3D infinite potential well. Use it when studying quantum confinement in nanostructures, quantum dots, or introductory quantum mechanics courses.
About this calculator
The particle-in-a-box (infinite square well) is a foundational quantum mechanics model. For a particle of mass m confined to a 1D box of length L, the allowed energies are Eₙ = (n²π²ℏ²) / (2mL²), where n = 1, 2, 3, … is the quantum number and ℏ = 1.055×10⁻³⁴ J·s. The energy grows as n², so higher states are widely spaced. In a 2D square box (sides L), the energy is E = (nx²+ny²)π²ℏ²/(2mL²), and in a 3D cube E = (nx²+ny²+nz²)π²ℏ²/(2mL²). Zero-point energy (n=1 minimum) is non-zero, a direct consequence of the uncertainty principle: confining the particle to length L implies a minimum momentum uncertainty ℏ/(2L). This model approximates electrons in conjugated molecules, conduction electrons in thin films, and quantum dots.
How to use
Find the ground-state energy (n = 1) of an electron (m = 9.109×10⁻³¹ kg) in a 1D box of length L = 1 nm = 1×10⁻⁹ m. Apply E₁ = (1²×π²×(1.055×10⁻³⁴)²) / (2 × 9.109×10⁻³¹ × (1×10⁻⁹)²). Numerator: π² × 1.113×10⁻⁶⁸ = 9.869 × 1.113×10⁻⁶⁸ = 1.098×10⁻⁶⁷ J²·s²/m². Denominator: 2 × 9.109×10⁻³¹ × 10⁻¹⁸ = 1.822×10⁻⁴⁸ kg·m². E₁ = 1.098×10⁻⁶⁷ / 1.822×10⁻⁴⁸ ≈ 6.02×10⁻²⁰ J ≈ 0.376 eV. Enter dimension = 1D, length = 1e-9 m, n = 1, mass = 9.109e-31 kg.
Frequently asked questions
Why does the particle in a box have a non-zero minimum energy?
The ground-state energy E₁ = π²ℏ²/(2mL²) is always positive and non-zero — the particle can never be completely at rest inside the box. This zero-point energy is a direct consequence of Heisenberg's uncertainty principle: confining the particle to a region of size L creates a momentum uncertainty Δp ≥ ℏ/(2L), which corresponds to a minimum kinetic energy. As the box shrinks, zero-point energy increases as 1/L², which is why electrons in small quantum dots have higher energy gaps and emit shorter-wavelength (bluer) light.
How does the particle-in-a-box model apply to real quantum dots and nanomaterials?
Semiconductor quantum dots confine electrons and holes in all three spatial dimensions on a scale of 2–10 nm. The particle-in-a-box model (adapted with the effective mass of the carrier and finite well depth) predicts that the optical band gap increases as the dot size decreases — the quantum confinement effect. This is why CdSe quantum dots can be tuned from red to blue simply by changing their diameter. The model also applies to π-electrons in conjugated polymers, conduction electrons in thin metallic films, and neutrons confined within atomic nuclei.
What happens to the energy levels when the box size is doubled?
Since Eₙ ∝ 1/L², doubling the box length L reduces every energy level by a factor of four. For example, the ground-state energy drops from E₁ to E₁/4. The spacing between consecutive levels also shrinks, meaning the energy spectrum becomes denser as the box grows. In the limit of a very large box (macroscopic object), the levels are so closely spaced they appear continuous — reproducing classical behaviour. This is the correspondence principle: quantum mechanics must reproduce classical results at large scales.