Particle in a Box Calculator
Enter mass, box length and quantum number to compute the energy levels of a 1D particle in a box E_n = n²·h²/(8·m·L²).
Input Data
Results
At a glance:The one-dimensional infinite potential well ('particle in a box') is the simplest quantum system: a particle of mass m confined between x=0 and x=L by infinitely high walls. Solving the Schrödinger equation gives quantized energies E_n = n²·h²/(8·m·L²) (n = 1,2,3...), wavefunctions sin(nπx/L), and a ground state E_1 = h²/(8mL²) (zero-point energy — the particle is never at rest). The spacing grows with n², so levels get farther apart as n increases. Transitions emit/absorb photons of energy ΔE = E_n − E_m. This model explains the electronic spectra of conjugated dyes and quantum dots (where L sets the confinement). This tool returns E_n, E_1 and a transition energy from m, L, n.
Formula
E_n = n²·h²/(8·m·L²)
Ground: E_1 = h²/(8·m·L²)
ΔE = E_n − E_m
$$E_n = \frac{n^2 h^2}{8mL^2}, \quad E_1 = \frac{h^2}{8mL^2}, \quad \Delta E_{n \to n+1} = (2n+1)E_1, \quad \psi_n = \sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}$$How to Use
- Enter the particle mass m (kg).
- Enter the box length L (m) and quantum number n.
- The calculator returns E_n, E_1 and a transition energy.
Case Studies
Electron in anano box
m = 9.1e-31 kg, L = 1e-9 m, n = 1.
E_1 = (6.626e-34)²/(8×9.1e-31×1e-18) ≈ 6.0e-20 J ≈ 0.37 eV.
Transition 1→2: ΔE = 3·E_1 ≈ 1.1 eV (visible/IR).
FAQ
Why is there a ground-state (zero-point) energy?
The uncertainty principle forbids the particle from having both zero momentum and a known position in the box; the lowest state still has kinetic energy E_1 > 0.
How does box size affect energy?
E_n ∝ 1/L² — confining the particle to a smaller box raises all energy levels, which is why quantum dots change color with size.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.