What is the energy of a particle in a box?

For an infinite square well of width L, Eₙ = n²h²/(8mL²) with n = 1, 2, 3 and so on. For an electron in a 1 nm box the ground state is 0.376 eV, and the levels go up as n², so n = 2 is 1.50 eV and n = 3 is 3.38 eV.

Why is the lowest energy of a particle in a box not zero?

A wave that must vanish at both walls needs at least half a wavelength across the box, so its wavelength is at most 2L and its momentum at least h/(2L). That leftover kinetic energy is the zero-point energy, and it fits the uncertainty principle: confining the particle forces a spread of momentum.

What is quantum tunnelling?

A particle meets a barrier higher than its energy. Classically it must bounce back, but the wavefunction decays inside the barrier instead of stopping at its edge, so if the barrier is thin some wave reaches the far side. The particle then has a small probability of being found there. Tunnelling explains alpha decay, the scanning tunnelling microscope and flash memory.

How does the transmission depend on barrier width?

For a thick or high barrier T is roughly 16(E/V₀)(1 − E/V₀) e^(−2κa), so each extra 1/(2κ) of width cuts the transmission by a factor of e. For an electron 2 eV below the top, κ is about 7.2 per nm, so adding 0.1 nm of barrier cuts T by about four times.

← All simulations

Particle in a box and tunnelling

Trap an electron in a tiny box and it can only have certain energies. Fire one at a barrier it cannot climb and some of it still gets through.

Probability density |ψ|²

Readouts

What's happening

In quantum mechanics a particle is described by a wavefunction ψ, and |ψ|² is the probability per unit length of finding it there. In an infinite square well of width L the wave must vanish at both walls, so only whole numbers of half-wavelengths fit: λ = 2L/n. Through de Broglie's p = h/λ that fixes the energy, Eₙ = n²h²/(8mL²), so the levels grow as 1, 4, 9, 16 and the lowest energy is not zero. State n has n − 1 nodes inside the box. A mix of two states is not stationary: the probability sloshes from side to side with period h/(E₂ − E₁). At a barrier of height V₀ higher than the particle's energy E, the wave does not stop dead at the wall but decays inside as e^−κx, with κ = √(2m(V₀ − E))/ħ. If the barrier is thin, some wave is left on the far side: that is tunnelling, and the transmission falls roughly as e^−2κa with barrier width a. Above the barrier the particle can still reflect, except at resonances where a whole number of half-wavelengths fits across the barrier and T = 1.

Eₙ = n²h² / (8mL²)ψₙ = √(2/L) sin(nπx/L)κ = √(2m(V₀ − E)) / ħT ≈ 16(E/V₀)(1 − E/V₀) e^−2κa

First-year university quantum physics: the infinite square well, normalisation, superposition, and transmission through a rectangular barrier. A-Level Physics (AQA, OCR): energy levels and the de Broglie wavelength.

Work through the numbers with Physics Formulas and Scientific Calculator.

Challenge

Predict first: an electron is trapped in a box 1.0 nm wide. What is its ground-state energy in eV? If you make the box twice as wide, what happens to that energy? Use E = h²/(8mL²) and check with the readout.

FAQ

What is the energy of a particle in a box?: For an infinite square well of width L, Eₙ = n²h²/(8mL²) with n = 1, 2, 3 and so on. For an electron in a 1 nm box the ground state is 0.376 eV, and the levels go up as n², so n = 2 is 1.50 eV and n = 3 is 3.38 eV.
Why is the lowest energy of a particle in a box not zero?: A wave that must vanish at both walls needs at least half a wavelength across the box, so its wavelength is at most 2L and its momentum at least h/(2L). That leftover kinetic energy is the zero-point energy, and it fits the uncertainty principle: confining the particle forces a spread of momentum.
What is quantum tunnelling?: A particle meets a barrier higher than its energy. Classically it must bounce back, but the wavefunction decays inside the barrier instead of stopping at its edge, so if the barrier is thin some wave reaches the far side. The particle then has a small probability of being found there. Tunnelling explains alpha decay, the scanning tunnelling microscope and flash memory.
How does the transmission depend on barrier width?: For a thick or high barrier T is roughly 16(E/V₀)(1 − E/V₀) e^−2κa, so each extra 1/(2κ) of width cuts the transmission by a factor of e. For an electron 2 eV below the top, κ is about 7.2 per nm, so adding 0.1 nm of barrier cuts T by about four times.