Particle in a Box: Energy Level E_n
Enter quantum number n, mass in kg and well width L in meters to get the energy in joules from E_n = n^2 h^2 / (8 m L^2), with h = 6.626e-34.
E_n (J)
—
Particle in a box: E_n = n²·h²/(8mL²)
Trap a particle of mass m inside a 1D infinite well of width L and its energy can only take certain values, E_n = n²·h²/(8mL²), where n runs through 1, 2, 3 and so on. There is no n = 0. What surprises most people is that the ground state E_1 sits above zero. That floor is the zero-point energy, and it follows straight from Heisenberg's uncertainty principle, since pinning down Δx forces Δp above zero. Because the levels go as n², the gaps between them widen the higher you climb. Take an electron (m = 9.11·10⁻³¹ kg) in a box of L = 1 nm: its E_1 lands near 6.0·10⁻²⁰ J, roughly 0.376 eV. Shrink L tenfold and E jumps by a factor of 100. That sensitivity to size is exactly why quantization only shows up at the nanoscale. Make the box big and the levels pack together so tightly they blur into a continuum.
Applications
You meet this model in quantum wells (QW) inside semiconductor lasers and high-brightness LEDs, in quantum dots (QLED TVs, fluorescent biological markers), and in conjugated polymers, where the free-electron picture describes the π-electron system. It also shows up in molecular orbital theory. And it opens the first chapter of nearly every quantum mechanics textbook, mostly because the math actually has a clean, exact solution.
FAQ
Why can't n = 0? Plug in n = 0 and you get ψ ≡ 0 everywhere, which describes no particle at all. Between the boundary conditions ψ(0) = ψ(L) = 0 and the need to normalize, n has to be at least 1.
What does zero-point energy mean physically? Cool the system down to T = 0 K and the particle still won't sit still. Being confined within L forces a nonzero minimum kinetic energy, so E_1 stays above 0.
Does this work in 3D? It does. The energy turns into a sum, E = (h²/8m)(n_x²/L_x² + n_y²/L_y² + n_z²/L_z²), so now you carry three quantum numbers. When the box is a cube, different combinations can land on the same energy, giving you degeneracies.
Related Tools
Wave Group Velocity
Calculate the group velocity of an ocean wave in deep water, c_g = g·T ÷ (4π), from the period T (s). The result, in m/s, is the speed at which the wave energy (and the 'envelope' of a wave group) propagates — exactly half the celerity (phase velocity) in deep water. This difference explains a curious phenomenon: within a wave group, individual crests appear at the rear, advance through the group (faster than it) and disappear at the front. The group velocity is what matters for energy transport and predicting swell arrival at the coast. Enter the wave period.
Kinetic Energy Calculator
Compute kinetic energy KE = ½ m v² from mass (kg) and velocity (m/s). Result in joules.
Hydraulic Jump Energy Loss
Calculate the specific energy dissipated in a hydraulic jump, ΔE = (y₂ − y₁)³ ÷ (4·y₁·y₂), from the upstream y₁ (supercritical) and downstream y₂ (subcritical) sequent depths. The hydraulic jump is one of the most efficient energy dissipators in hydraulics: intense turbulence in the transition converts kinetic energy to heat and sound, removing excess flow energy. This head loss ΔE is exactly what is sought downstream of spillways, gates and bottom outlets — water arrives with very high energy (able to scour the riverbed and undermine the structure), and the stilling basin induces the jump to 'burn' that energy in a controlled way. The higher the incoming Froude number, the greater the dissipated fraction — jumps with Fr > 9 dissipate up to 85%. Enter the upstream and downstream sequent depths.
Critical Depth in Rectangular Channel
Calculate the critical depth of a rectangular channel, y_c = (q² ÷ g)^(1/3), from the unit discharge q (flow per unit width, m³/s/m) and gravity g. Critical depth is the depth at which specific energy is minimum, marking the boundary between the two open-flow regimes: above it the flow is subcritical (slow, deep, Fr < 1, downstream-controlled); below, supercritical (fast, shallow, Fr > 1, upstream-controlled); exactly at it, Fr = 1. Critical depth is central to channel and structure hydraulics: it defines the control section at spillways, weirs and flumes (Parshall), where flow passes through the critical regime stably and the stage-discharge relation is unique — allowing flow measurement from head. It also determines whether a hydraulic jump can form and guides water-surface profiles. Enter the channel's unit discharge.
Settling Velocity (Stokes)
Calculate the settling (terminal) velocity of a particle in laminar regime by Stokes' Law, v_s = g·d²·(ρ_s − ρ_w) ÷ (18·μ), from the particle diameter d (m), the solids ρ_s and water ρ_w densities (kg/m³) and the fluid dynamic viscosity μ (Pa·s). The settling velocity is the speed at which an isolated particle SINKS in a still fluid, when weight (minus buoyancy) balances drag. Stokes' Law (1851) holds for the LAMINAR regime (small particles, particle Reynolds < ~1) — fine sand, silt, clay — and has the remarkable property that velocity grows with the SQUARE of diameter: particles twice as large sink four times faster. This calculation is fundamental in many fields: particle settling and separation (settling tanks, thickeners, water and effluent clarifiers), grain-size classification by sedimentation (pipette or hydrometer test), sediment transport in rivers and reservoir deposition, and hydraulic transport (the particle settling velocity sets the critical deposition velocity in the pipe). For large particles (higher Reynolds), Stokes' Law fails and Newton's terminal velocity (turbulent regime) is used. Enter the particle diameter, the densities and the viscosity.
Grinding Energy (Bond Work Index)
Calculate the specific comminution (grinding/crushing) energy by Bond's law, W = 10 × Wi × (1/√P₈₀ − 1/√F₈₀), from the ore's Bond work index Wi (kWh/t), and the particle sizes passing 80% of the product (P₈₀) and feed (F₈₀), in micrometers. The result, in kWh per tonne, is the energy needed to reduce the ore from feed to product size. Comminution is mining's largest energy consumer (up to 50% of the plant). The Wi index characterizes the ore's resistance to fragmentation. It is the basis for sizing mills and energy consumption. Enter the Wi, P₈₀ and F₈₀.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.