Particle in a Box

Q: Why does the energy grow as $n^2$?

For level $n$ the wave number $k$, which measures how many radians the wave's phase turns per meter, is $k = \frac{n\pi}{L}$. The energy depends on $k^2$, and $k^2$ carries the factor $n^2$.

In one sentence: A particle trapped between two walls can have only certain energies, like steps on a ladder, and the narrower the box, the higher those steps are.

$E_n = \frac{n^2 h^2}{8 m L^2}, \qquad n = 1, 2, 3, \dots$

Read it out loud: “E sub n equals n squared times h squared, all over 8 times m times L squared, with n equal to 1, 2, 3 and so on. E sub n is the energy of level n, in joules. n is the level number, starting at 1 and counting up by whole numbers. h is Planck's constant, m is the particle's mass in kilograms, and L is the width of the box in meters.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Powers of ten, like $10^{-9}$ and $10^{-34}$: Many numbers in this formula are tiny, so they use powers of ten. A negative exponent means divide by ten that many times. So $10^{-9}$ is one billionth, and a 1.0 nm box has width $L = 1.0 \times 10^{-9}$ m. Planck’s constant is $6.62607015 \times 10^{-34}$ J s, which has 33 zeros after the decimal point before its first digit. To multiply powers of ten, add the exponents: $10^{-9} \times 10^{-9} = 10^{-18}$ .
Energy levels are quantized: A quantum particle in a box cannot have any energy it likes. Its energies come in a list of allowed values, called energy levels. The particle is found at one level at a time, and it cannot sit between two levels. Physicists say the energy is quantized, which means it comes in fixed steps like the rungs of a ladder instead of changing smoothly.
Standing waves, half-waves and the walls: The particle’s wave must have zero height at both walls, because the particle cannot be outside the box. Only some wave patterns fit between the walls. The simplest pattern is one half-wave, a single hump from wall to wall. The next pattern has two half-waves, then three, and so on. A wave that stays in place while its height changes is called a standing wave.
Joules, electron volts and the unit J s: The joule (J) is the unit of energy. One joule is roughly the energy needed to lift a small apple one meter. An electron volt (eV) is a much smaller energy, $1.602176634 \times 10^{-19}$ J, which is the energy an electron gains when pushed through one volt. The unit J s, joule seconds, belongs to Planck’s constant $h$ . Multiplying joule seconds by one per second gives joules, which is a handy check.
Mass in kilograms, length in meters and the nanometer: Mass is the amount of matter, and its unit is the kilogram (kg). An electron has a mass of $9.1093837 \times 10^{-31}$ kg, which is extremely small. Length is measured in meters (m). A nanometer (nm) is one billionth of a meter, so $1\ \text{nm} = 10^{-9}$ m. A box 1.0 nm wide is small enough to fit only a few atoms across.
Momentum $p$ and the de Broglie wavelength: Momentum, written $p$ , is mass multiplied by speed. A moving particle has a wavelength given by $\lambda = h/p$ , which is called the de Broglie wavelength. A bigger momentum gives a shorter wavelength, so a faster electron has a shorter wave. This link between particles and waves is the reason the box allows only certain half-wave patterns.

What each symbol means

Symbol	Meaning
$E_n$	The energy of level $n$ , in joules (J). Atoms and molecules are often described in electron volts (eV) instead, where $1\ \text{eV} = 1.602 \times 10^{-19}$ J. The energy is always positive, and each level has one fixed value.
$n$	The level number, a whole number starting at 1, so $n = 1, 2, 3, \dots$ . The dots mean the list goes on, with the next whole number each time. It counts the half-waves that fit in the box, and it has no unit.
$h$	Planck’s constant, $6.62607015 \times 10^{-34}$ J s, read as joule seconds. Its value is exact by definition in the SI system, which means it is fixed by definition and not measured with error. It links energy to frequency for light and for matter.
$m$	The mass of the particle, in kilograms (kg). For an electron, $m = 9.1093837 \times 10^{-31}$ kg, a very small mass. A heavier particle has lower energy levels, because the mass sits in the bottom of the formula.
$L$	The width of the box, the distance between the two walls, in meters (m). The particle stays between the walls. A width of 1.0 nm equals $1.0 \times 10^{-9}$ m. Halving $L$ makes every energy four times larger, because $L$ is squared.

The formula piece by piece

$n$: Start with the level number $n$ , a whole number 1, 2, 3 and so on. It counts the half-waves that fit inside the box. Level 1 has one half-wave, level 2 has two, and level 3 has three.
$n^2$: Square it, so $n^2$ means $n$ times $n$ . Level 2 gives $n^2 = 4$ , and level 3 gives $n^2 = 9$ . The energies grow faster than the level number, so the steps between levels get bigger as you go up the ladder.
$h^2$: Next, $h^2$ is Planck’s constant multiplied by itself. It is extremely small, about $4.39 \times 10^{-67}$ J $^2$ s $^2$ . Quantum constants this small make the energy steps tiny for everyday objects, which is why we do not notice the steps for a baseball or a car.
$8 m L^2$: The bottom of the fraction is $8 m L^2$ , which is 8 times the mass times the width squared. A heavier particle or a wider box makes this bottom bigger, which makes the energy smaller. Because $L$ is squared, halving the width makes the bottom four times smaller and the energy four times bigger.
$\frac{n^2 h^2}{8 m L^2}$: Put the top over the bottom: $n^2 h^2$ divided by $8 m L^2$ . The units work out, because joule seconds squared divided by kilogram meters squared equals joules. So this fraction is an energy.
$E_n = \frac{n^2 h^2}{8 m L^2}, \qquad n = 1, 2, 3, \dots$: The whole formula in one sentence: the energy of level $n$ in a box equals $n$ squared times $h$ squared, divided by eight times the particle’s mass times the box width squared, for $n = 1, 2, 3$ and so on.

What the formula is doing, and why

A particle trapped between two walls is one of the first problems in quantum mechanics. What goes in is the particle’s mass, the box width and the level number. What comes out is the energy of that level. The wave of the particle must be zero at both walls. Only certain wave patterns fit between them, and each pattern has one energy, so the particle can have only certain energies.

Why $n$ , and why $n^2$ ? Level $n$ has $n$ half-waves in the box, so its wavelength is $\lambda = 2L/n$ . A particle with wavelength $\lambda$ has momentum $p = h/\lambda = nh/(2L)$ . A moving particle’s energy is $p^2/(2m)$ . Putting the momentum in gives $E = \frac{n^2 h^2}{8 m L^2}$ . That is why energy grows with the square of $n$ .

Why the box width matters so much. The energy depends on $L^2$ in the bottom line, so halving the width makes every energy four times bigger. Shrinking the box pushes the levels up quickly. A heavier particle has lower energies, because $m$ is in the bottom line too. This matters at the nanometer scale, where electron energies are in the electron-volt range. In a box one meter wide, the levels are packed so tightly that they look like a smooth range of energies.

Why is the lowest energy above zero? The lowest level, $n = 1$ , is not zero. A wave that must vanish at both walls cannot be flat, so it has to bend between the walls. Bending means the particle carries some kinetic energy, which is energy of motion. A particle with zero energy would have to be perfectly still and flat, and that is impossible in a box.

Where does it apply? This is an idealized model with walls that are infinitely high and a particle that moves freely between them. Real walls are finite, so the wave leaks into them a little and the levels shift. The formula is exact for the ideal box, and it is a good first picture of electrons trapped in tiny regions.

An everyday picture

Think of a ladder whose rungs spread out as you climb. The particle can stand only on the rungs, never between them. The first rung sits low, the second is four times as high above the floor, and the third is nine times as high. A narrower box is like a ladder with wider gaps between the rungs, so every rung is higher and the steps between them are bigger.

Worked example: An electron in a 1.0 nm box

An electron, with mass $m = 9.1093837 \times 10^{-31}$ kg, is trapped in a box of width $L = 1.0$ nm $= 1.0 \times 10^{-9}$ m. We find the energies of the first three levels in electron volts (eV), and the energy needed to move from level 1 to level 2. The box width is chosen for easy numbers, so this is an idealized example.

Write the level 1 formula with the numbers. For level 1, $n = 1$ , so the formula becomes $E_1 = \frac{h^2}{8 m L^2}$ . Put in $h = 6.62607015 \times 10^{-34}$ J s, $m = 9.1093837 \times 10^{-31}$ kg and $L = 1.0 \times 10^{-9}$ m. Squaring $L$ means multiplying its power of ten by two, so $(10^{-9})^2 = 10^{-18}$ .
Square the top and multiply the bottom. The top, $h^2$ , has front number $6.62607015^2 \approx 43.904$ , and the exponent doubles to $-68$ . Tidying gives $h^2 \approx 4.3904 \times 10^{-67}$ J $^2$ s $^2$ . The bottom is $8 \times 9.1093837 \times 10^{-31} \times 10^{-18} \approx 7.2875 \times 10^{-48}$ kg m $^2$ .
Divide to get the energy in joules. Dividing the top by the bottom gives $\frac{4.3904 \times 10^{-67}}{7.2875 \times 10^{-48}} \approx 0.60247 \times 10^{-19}$ . The powers of ten subtract: $-67 - (-48) = -19$ . Moving the decimal point one place to the right gives $E_1 \approx 6.0247 \times 10^{-20}$ J, about $6.02 \times 10^{-20}$ J.
Convert to electron volts. One electron volt is $1.602176634 \times 10^{-19}$ J, so divide: $\frac{6.0247 \times 10^{-20}}{1.602177 \times 10^{-19}} \approx 0.37603$ eV. The power of ten in the bottom is one bigger, so the result shrinks by a factor of ten, and the front numbers give $6.0247 / 1.602177 \approx 3.7603$ . So $E_1 \approx 0.376$ eV.
Find level 2 and level 3 by squaring the level number. Level 2 has $n^2 = 4$ , so $E_2 = 4E_1 = 4 \times 0.37603 \approx 1.504$ eV. Level 3 has $n^2 = 9$ , so $E_3 = 9E_1 = 9 \times 0.37603 \approx 3.384$ eV. The levels rise much faster than the level number does.
Find the energy needed to move from level 1 to level 2. The gap is $E_2 - E_1 = 4E_1 - E_1 = 3E_1 = 3 \times 0.37603 \approx 1.128$ eV. An electron that absorbs about 1.13 eV, for example from a photon of that energy, can jump up one level.

Answer: The first three levels of this electron are 0.376 eV, 1.50 eV and 3.38 eV. Moving from level 1 to level 2 takes about 1.13 eV. The lowest level sits above zero, because a wave that must vanish at both walls cannot be flat.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Leaving the width in nanometers. Convert 1.0 nm to $1.0 \times 10^{-9}$ m before squaring. Using $L = 1.0$ as if it were meters makes the energy $10^{18}$ times too small.
Assuming the lowest energy can be zero. Level $n = 1$ is above zero, because a wave that must vanish at both walls cannot be flat.
Starting the count at $n = 0$ . That would give zero energy and a wave that is zero everywhere, which means no particle. The level numbers start at 1.
Doubling the energy when $n$ doubles. The energy depends on $n^2$ , so $E_2 = 4E_1$ and $E_3 = 9E_1$ , not $2E_1$ and $3E_1$ .

Where you will meet it

Quantum dots, tiny semiconductor crystals whose color depends on their size, because the size sets how far apart the trapped electron levels are.
Quantum well lasers and LEDs, which use very thin semiconductor layers to trap electrons in one direction, with energy levels set by the same idea.
Estimating the energies of electrons along the long chains of carbon atoms in some dye molecules, using a one-dimensional box as a simple model.

Questions people ask

Why does the level count start at 1 and not 0?

With $n = 0$ the wave would be zero everywhere, so there would be no particle in the box. The lowest allowed wave has one half-wave, which is $n = 1$ .

Why does the energy grow as $n^2$?

For level $n$ the wave number $k$ , which measures how many radians the wave’s phase turns per meter, is $k = \frac{n\pi}{L}$ . The energy depends on $k^2$ , and $k^2$ carries the factor $n^2$ .

Is a real electron trapped in a box like this?

Not exactly. Real walls are finite, so the levels shift a little and the wave leaks into the walls. The ideal box is the simplest model that shows why energy comes in discrete levels.

Schrödinger Equation: The Schrödinger equation says how a quantum particle’s wave changes from moment to moment, the way Newton’s laws say how a moving ball changes its position.
Quantum Harmonic Oscillator: A tiny particle on a spring can have only certain energies, spaced evenly like the steps of a staircase, and even its lowest step is above zero.
Hydrogen Energy Levels: An electron in a hydrogen atom can sit only on certain energy steps, and when it jumps between steps it gives off or takes in light of one exact color.