Energy Levels of the Particle in a Box

Introduction

The boundary conditions of the one-dimensional box allow only waves with k = nπ/L. Turning that result into a formula for the energy gives one of the most widely used equations in quantum chemistry: Eₙ = n²h²/8mL² . It tells us how the energy grows with the quantum number, how it depends on the mass of the particle and on the size of the box, and how far apart the levels are. Those three dependences explain why quantum effects dominate for electrons in molecules but vanish for everyday objects.

Core explanation

Deriving the formula. Inside the box, E = ħ²k²/2m. Substituting k = nπ/L gives

Eₙ = ħ²n²π²/2mL²

Since ħ = h/2π, ħ²π² = h²/4, and so

Eₙ = n²h²/8mL², n = 1, 2, 3, …

All the energy is kinetic, because V = 0 inside the box.

Dependence on n. The energies are proportional to n². If the ground-state energy is E₁ = h²/8mL², the levels are E₁, 4E₁, 9E₁, 16E₁, … The ladder has rungs that get further apart as you climb.

Level spacing. The gap between neighbouring levels is

ΔE = Eₙ₊₁ − Eₙ = [(n + 1)² − n²]h²/8mL² = (2n + 1)h²/8mL²

The gap from n = 1 to 2 is 3E₁, from 2 to 3 is 5E₁, from 3 to 4 is 7E₁: the odd numbers.

Dependence on L. Energies are proportional to 1/L². Halving the box length quadruples every energy and every gap. Confinement raises energy: a tighter box forces shorter wavelengths, sharper curvature and more kinetic energy.

Dependence on m. Energies are proportional to 1/m. Light particles in small boxes have widely spaced levels; heavy particles in large boxes have levels so close that they appear continuous.

Putting in numbers. For an electron (m = 9.109 × 10⁻³¹ kg) in a box of length 1.0 nm:

E₁ = (6.626 × 10⁻³⁴)² / (8 × 9.109 × 10⁻³¹ × (1.0 × 10⁻⁹)²) ≈ 6.0 × 10⁻²⁰ J, about 0.38 eV

This is comparable to chemical energies, so quantisation is decisive. By contrast, a nitrogen molecule (about 4.7 × 10⁻²⁶ kg) in a 1 cm box has E₁ of order 10⁻³⁸ J, vastly smaller than the thermal energy kT ≈ 4 × 10⁻²¹ J at room temperature, so its translational levels form an effective continuum.

Spectroscopy. A photon can promote the particle from level n to n + 1 if hν = ΔE. Because ΔE ∝ 1/L², longer boxes absorb longer wavelengths — the basis of the free-electron model of dye colours.

Formulae

Eₙ = n²h²/8mL². ΔE(n → n + 1) = (2n + 1)h²/8mL². Absorption wavelength λ = hc/ΔE. Constants: h = 6.626 × 10⁻³⁴ J s, mₑ = 9.109 × 10⁻³¹ kg, c = 2.998 × 10⁸ m s⁻¹.

Step-by-step reasoning

To predict a transition wavelength for a particle in a box:

1. Convert L to metres and identify the mass m in kg. 2. Calculate h²/8mL², which is E₁. 3. Find ΔE for the transition, for example 3E₁ for n = 1 → 2. 4. Convert to wavelength with λ = hc/ΔE. 5. Check the magnitude against the expected region of the spectrum.

Visual explanation

Draw horizontal lines inside the box at heights 1, 4, 9 and 16 units. The gaps grow as 3, 5 and 7 units. Now redraw the box twice as wide: every line drops to a quarter of its height, and the ladder is compressed towards the floor.

Real-world analogy

Think of a xylophone. Shorter bars ring at higher pitches, and within a single bar the overtones rise in a fixed pattern. Shrinking the box is like shortening the bar: every level moves up, and the pattern stays the same.

Real-world example

Cadmium selenide quantum dots a few nanometres across behave like electrons in tiny boxes. Smaller dots have larger level spacings and emit bluer light; larger dots emit redder light. Manufacturers tune the colour of display pixels simply by controlling particle size.

Why?

Why does energy grow as n² rather than n? The energy is kinetic, proportional to p², and p = h/λ with λ = 2L/n. Momentum grows linearly with n, so energy grows with its square.

Common misconception

"The levels are equally spaced, like a ladder with even rungs." That is true for the harmonic oscillator, not for the box. In the box the gaps grow as 2n + 1, so higher levels are further apart.

Worked example

Question: Calculate the wavelength of light absorbed when an electron in a 1.0 nm box is promoted from n = 1 to n = 2.

Reasoning: E₁ = 6.02 × 10⁻²⁰ J (from above). ΔE = E₂ − E₁ = 3E₁ = 1.81 × 10⁻¹⁹ J. λ = hc/ΔE = (6.626 × 10⁻³⁴ × 2.998 × 10⁸) / 1.81 × 10⁻¹⁹ = 1.10 × 10⁻⁶ m.

Answer: About 1100 nm, in the near infrared.

Quick check

1. If the length of a box is doubled, by what factor does the energy of each level change? Answer: Each energy falls to one quarter of its original value, because Eₙ is proportional to 1/L².

Exam focus

Learn Eₙ = n²h²/8mL² and the spacing (2n + 1)h²/8mL². Most errors come from units: L must be in metres and m in kilograms. Be ready to explain qualitatively why energies rise with n and fall with m and L.

Advanced insight

The ratio of level spacing to kT decides whether a motion must be treated quantum mechanically. For translation in macroscopic containers the spacing is tiny, so the translational partition function can be evaluated by replacing the sum over n with an integral, which yields the classical result that each translational degree of freedom contributes ½kT to the mean energy.

Summary

For a particle in a one-dimensional box, Eₙ = n²h²/8mL² with n = 1, 2, 3, … Energies rise as n², so adjacent gaps are (2n + 1)h²/8mL² and grow up the ladder. Energies fall as 1/m and 1/L²: light particles in small boxes have widely spaced levels that matter chemically, while heavy particles in large boxes behave classically.

Practice questions

1. Calculate E₁ for an electron in a box of length 0.20 nm. Answer: E₁ = (6.626 × 10⁻³⁴)² / (8 × 9.109 × 10⁻³¹ × (2.0 × 10⁻¹⁰)²) ≈ 1.5 × 10⁻¹⁸ J. 2. What is E₃ − E₂ in units of h²/8mL²? Answer: 9 − 4 = 5. 3. How does E₁ for a proton compare with E₁ for an electron in the same box? (mₚ/mₑ ≈ 1836) Answer: It is about 1836 times smaller, because E is inversely proportional to mass. 4. Which transition, 1 → 2 or 2 → 3, absorbs the longer wavelength, and why? Answer: 1 → 2, because its energy gap (3 units) is smaller than that of 2 → 3 (5 units), and wavelength is inversely proportional to energy.