De Broglie Wavelength

In one sentence: Every moving object has a wave that goes with it, and the heavier or faster the object moves, the shorter that wave is.

$\lambda = \frac{h}{p} = \frac{h}{m v}$

Read it out loud: “Lambda equals h divided by p, which is the same as h divided by m times v. Lambda is the Greek letter for wavelength, the distance from one wave crest to the next. The letter h is Planck's constant. The letter p is momentum, which is mass times speed. The letter m is mass, and the letter v is speed.”

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 $10^{-31}$ and $10^{-34}$: Very large and very small numbers are written as a short number times a power of ten. The small raised number, the exponent, says how many times to multiply by 10: $10^{3} = 1000$ . A minus sign in the exponent means divide by 10 instead, so $10^{-31}$ is a decimal point followed by 30 zeros and then the digit 1. Writing numbers this way is called scientific notation. When you multiply two numbers like this, you add their exponents, and when you divide, you subtract them.
Mass and the kilogram (kg): Mass tells how much matter an object contains. It also decides how hard the object is to speed up or slow down. The standard unit is the kilogram, written kg. A liter of water has a mass of about 1 kg. A baseball has a mass of 0.145 kg, which is 145 grams. An electron is far lighter, with a mass of $9.1 \times 10^{-31}$ kg, so tiny that scientists write it with a power of ten.
Speed and meters per second (m/s): Speed tells how far something travels in one second. It is measured in meters per second, written m/s. A brisk walk is about 1.4 m/s, and a baseball thrown hard is about 40 m/s, which is about 145 kilometers per hour. The electron in the example moves at one million meters per second. That is about 3.6 million kilometers per hour, but still only about one third of one percent of the speed of light.
Momentum $p$ and the unit kg m/s: Momentum is mass times speed, written $p = mv$ . It measures how much motion an object has, counting both how much stuff is moving and how fast it moves. A heavy object or a fast object has a lot of momentum. Momentum is measured in kilogram meters per second, written kg m/s. A 0.145 kg ball moving at 40 m/s has momentum $0.145 \times 40 = 5.8$ kg m/s.
Wavelength and the Greek letter lambda, $\lambda$: The Greek letter $\lambda$ , said LAM-duh, is the usual symbol for a wavelength. A wavelength is the distance from one wave crest to the next. Light waves have wavelengths of a few hundred nanometers, and a nanometer, written nm, is one billionth of a meter. Water waves in a pond can have wavelengths of several centimeters, and ocean swells can be many meters long.
Wave and matter wave: A wave is a repeating pattern that moves through space, like ripples on a pond or sound moving through air. A matter wave is the name Louis de Broglie gave to the wave that goes with a moving particle. The matter wave is not water or air moving. It is a mathematical wave that tells us where the particle is likely to be found, and its wavelength depends on the particle’s momentum.

What each symbol means

Symbol	Meaning
$\lambda$	The de Broglie wavelength: the distance from one crest of the matter wave to the next, measured in meters (m). For an electron moving at one million meters per second, $\lambda$ is about $7.3 \times 10^{-10}$ m, which is 0.73 nm. Smaller wavelengths mean shorter waves.
$h$	Planck’s constant, a fixed number of nature: $h = 6.62607015 \times 10^{-34}$ J s, with the unit joule seconds. The value is the same for every object. Because $h$ is so small, the wavelength of everyday objects is extremely tiny.
$p$	The momentum of the particle, measured in kilogram meters per second (kg m/s). It is the mass times the speed, $p = mv$ . For the electron in the example, $p = 9.1 \times 10^{-25}$ kg m/s. For a baseball thrown at 40 m/s, it is 5.8 kg m/s.
$m$	The mass of the particle, in kilograms (kg). Mass is the amount of matter in the object. The electron has a mass of $9.1 \times 10^{-31}$ kg, while a baseball has 0.145 kg, a little more than a small apple.
$v$	The speed of the particle, in meters per second (m/s). It says how far the particle travels in one second. The electron in the example moves at $1.0 \times 10^{6}$ m/s, and the baseball at 40 m/s, which is about 145 km/h.

The formula piece by piece

$m$: Start with the mass, the amount of matter in the particle. More mass means more momentum for the same speed, and that also means a shorter wavelength. Electrons have very little mass, which is why their waves are large enough to measure.
$v$: Next is the speed, how fast the particle moves. Faster motion means more momentum, which means a shorter wavelength. Mass and speed enter the formula only through their product, so a heavy slow object and a light fast object can have the same momentum.
$m v$: Multiply mass by speed to get momentum, $p = mv$ . Multiplying the two means both quantities count. The unit kilogram times meter per second is written kg m/s, and it is the amount of motion the particle carries.
$h$: Planck’s constant is the fixed conversion number linking a wave to a particle. Because it is divided by the momentum, a bigger momentum makes the wavelength smaller, and the constant sets the scale of the whole relation for every object.
$\frac{h}{p}$: Divide Planck’s constant by the momentum to get the wavelength. A bigger momentum means a smaller answer, so the wavelength shrinks as the momentum grows. Doubling the momentum cuts the wavelength in half.
$\lambda = \frac{h}{p} = \frac{h}{m v}$: The whole formula in one sentence: the wavelength of a moving particle equals Planck’s constant divided by the particle’s mass times its speed. Mass and speed go in, and the wavelength comes out.

What the formula is doing, and why

What goes in. The mass $m$ and the speed $v$ of a moving particle. What happens. Multiply them to get the momentum $p = mv$ , then divide Planck’s constant $h$ by $p$ . What comes out. The wavelength $\lambda$ , in meters. Every quantity in the formula is measured in a standard unit, so you can put in numbers for any moving object.

Where it came from. In 1924 Louis de Broglie proposed, in his doctoral thesis, that particles such as electrons also have a wave side, as light has a particle side. Electron waves were observed in 1927 by Clinton Davisson and Lester Germer, and independently by G. P. Thomson. The idea was a bold extension of the photon picture of light, and experiments confirmed it.

Why divide? A wave with a large momentum has a short wavelength, so its crests are packed closely together. A wave with a small momentum has a long wavelength, so its crests are spread out. The relation is an inverse one, which means that when one quantity goes up, the other goes down. Dividing $h$ by $p$ captures exactly that inverse relation.

Why is $h$ so tiny? Because Planck’s constant is so small, the wavelength $h/p$ is tiny unless the momentum is also tiny. Everyday objects have large momentum, so their wavelengths are far too short to detect. A light particle with a small momentum, such as an electron moving slowly, has a wavelength large enough to measure in an experiment.

When it applies and what it means. The formula applies to every moving particle. The wave shows its effects only when its wavelength is comparable to the size of the things it meets, such as the spacing between atoms in a crystal. In modern quantum mechanics the matter wave is part of the wave function, the quantity that predicts probabilities for the particle. It is not a ripple in some material.

An everyday picture

Think of a ping-pong ball and a bowling ball rolling at the same speed. The bowling ball carries far more momentum because it has far more mass, so its matter wave is far shorter. A wave shows its effects only when its wavelength is comparable to the things around it. For a bowling ball that never happens. For an electron inside an atom, which is about a tenth to a third of a nanometer across, it does.

Worked example: An electron and a baseball

An electron with mass $m_e = 9.1093837 \times 10^{-31}$ kg moves at $v = 1.0 \times 10^{6}$ m/s. A baseball with mass 0.145 kg is thrown at $v = 40$ m/s, a fast pitch of about 145 km/h. We want the matter wavelength of each, using Planck’s constant $h = 6.62607015 \times 10^{-34}$ J s.

Find the electron’s momentum. Momentum is mass times speed, so we multiply $9.1093837 \times 10^{-31}$ kg by $1.0 \times 10^{6}$ m/s. The front numbers multiply to $9.1093837$ . For the powers of ten we add the exponents, $-31 + 6 = -25$ , so $p = 9.1093837 \times 10^{-25}$ kg m/s.
Divide Planck’s constant by the momentum. The wavelength is $\lambda = h / p = \frac{6.62607015 \times 10^{-34}}{9.1093837 \times 10^{-25}}$ . The front numbers divide to about $0.7274$ . For the powers of ten we subtract the exponents, $-34 - (-25) = -9$ , so $\lambda \approx 0.7274 \times 10^{-9}$ m.
Put the answer in a friendlier unit. A nanometer is $10^{-9}$ m, so $0.7274 \times 10^{-9}$ m is the same as 0.7274 nm. The electron’s wavelength is therefore about 0.73 nanometers, a size that is easy to compare with atoms.
Compare the electron’s wave with an atom. A typical atom is roughly 0.1 to 0.3 nm across. The electron’s wavelength is therefore two to seven times the width of one atom. That is why wave behavior matters for electrons inside atoms and in electron microscopes.
Find the baseball’s momentum. Multiply mass by speed: $0.145\ \text{kg} \times 40\ \text{m/s} = 5.8$ kg m/s. The mass must be in kilograms and the speed in meters per second, which is why the 145 grams was first changed to 0.145 kg.
Find the baseball’s wavelength. Divide Planck’s constant by the momentum: $\lambda = \frac{6.62607015 \times 10^{-34}}{5.8} \approx 1.14 \times 10^{-34}$ m. The exponent stays at $-34$ , because we divided by 5.8, a plain number with no power of ten.
Compare the baseball’s wave with an atom. Take an atom to be about $10^{-10}$ m across. Dividing gives $\frac{1.14 \times 10^{-34}}{10^{-10}} \approx 1.14 \times 10^{-24}$ , so the baseball’s wave is about $10^{-24}$ times the width of an atom. Written out, $10^{-24}$ is a decimal point followed by 23 zeros and then a 1, far too small for any experiment to see.

Answer: The electron’s wave is about 0.73 nm long, which is two to seven times the width of one atom, so wave behavior matters for electrons in atoms. The baseball’s wave is about $1.14 \times 10^{-34}$ m long, far too short to ever measure. The same formula gives very different results because the wave matters only when the mass is tiny.

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

Common mistakes

Using speed where momentum belongs. The formula needs $p = mv$ , so the mass matters too. Leaving the mass out gives a wrong answer, off by a factor equal to the mass.
Using grams instead of kilograms. Planck’s constant is in SI units, so 145 g must become 0.145 kg before you divide.
Treating the wavelength as the size of the particle. The wavelength describes the wave that goes with the particle. It is not the particle’s radius, and it can be far larger or far smaller than the particle itself.
Assuming everyday objects show wave effects. A wavelength matters only when it is comparable to the size of the things the wave meets, which almost never happens for everyday objects.

Where you will meet it

Electron diffraction experiments, which confirmed the wave nature of electrons in 1927.
Electron microscopes, which use fast electrons with wavelengths of a few picometers. A picometer is a trillionth of a meter, so the electron waves are much smaller than an atom.
Neutron scattering, where neutrons with wavelengths close to the spacing between atoms reveal the structure of crystals.
Quantum mechanics textbooks, where a particle confined to one small region is described by a bundle of waves with different wavelengths.

Questions people ask

Does a baseball really have a wavelength?

Yes. The formula gives every moving object a de Broglie wavelength. For a baseball it is far too small to measure, so in daily life it has no effect that anyone can notice.

Is the electron a wave or a particle?

In experiments an electron is detected as a whole particle, one hit at a time. But when many electrons pass through a crystal or through two slits, their pattern builds up according to wave rules. Both pictures are useful, and each describes a different side of the same measurements.

Why is the wavelength so tiny for most objects?

Because $h = 6.6 \times 10^{-34}$ J s is so small, the wavelength $h/p$ is tiny unless the momentum $p$ is tiny too. Small masses moving slowly have small momentum, which is why they show wave behavior.

Heisenberg Uncertainty Principle: The more tightly you pin down where a tiny particle is, the more its speed must spread out, and no instrument can beat that trade-off.
Planck-Einstein Relation (E = hf): E = hf says each packet of light carries energy in proportion to its frequency: higher frequency means more energy per packet, and brightness does not change it.
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.