{"id":686,"date":"2026-07-31T23:00:04","date_gmt":"2026-07-31T23:00:04","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=686"},"modified":"2026-08-24T13:03:54","modified_gmt":"2026-08-24T13:03:54","slug":"de-broglie-wavelength","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/modern-physics\/de-broglie-wavelength\/","title":{"rendered":"De Broglie Wavelength: Formula, Examples and Worked Problems"},"content":{"rendered":"\n
Definition<\/div>

\nThe de Broglie wavelength is the wavelength associated with any moving particle, equal to Planck’s constant divided by the particle’s momentum (\u03bb = h \/ mv). Every object with mass and velocity has one. It is large enough to measure for electrons and neutrons, but unimaginably small for everyday objects like a cricket ball.\n<\/p><\/div>\n\n

Fire a stream of electrons at a thin crystal and they do something no particle should do. Instead of arriving as a scatter of dots, they pile up in bright rings and dark gaps — the unmistakable signature of a wave passing through a grating.<\/p>\n\n

That pattern is not a quirk of electrons. It is a property of everything that moves, and a young French PhD student worked out the rule for it in 1924 with one short equation.<\/p>\n\n

What Is the De Broglie Wavelength?<\/h2>\n\n

The de Broglie wavelength is the wavelength a moving particle behaves as if it has, found by dividing Planck’s constant by the particle’s momentum. Louis de Broglie proposed it in his 1924 doctoral thesis, and it applies to electrons, neutrons, atoms, molecules and, in principle, you.<\/p>\n\n

Physics already accepted that light — obviously a wave — sometimes behaves like a stream of particles. De Broglie asked the obvious question in reverse. If waves can act like particles, why shouldn’t particles act like waves?<\/p>\n\n

The idea sounds like a philosophical flourish. It isn’t. It makes a hard numerical prediction: give a particle a momentum, and its wavelength is fixed. Nothing is left to interpretation.<\/p>\n\n

The De Broglie Wavelength Formula<\/h2>\n\n

The de Broglie wavelength formula is Planck’s constant divided by momentum, written as lambda equals h over mv.<\/p>\n\n

\u03bb = h \/ (m\u00b7v)<\/div>\n\n

Because momentum p = m\u00b7v, the same equation is often written more compactly:<\/p>\n\n

\u03bb = h \/ p<\/div>\n\n
\n\n\n\n\n
Symbol<\/th>\nQuantity<\/th>\nSI unit<\/th>\n<\/tr>\n<\/thead>\n
\u03bb<\/strong><\/td>De Broglie wavelength<\/td>metre (m)<\/td><\/tr>\n
h<\/strong><\/td>Planck constant = 6.62607015 × 10-34<\/sup> (exact by definition)<\/td>joule second (J·s)<\/td><\/tr>\n
m<\/strong><\/td>Mass of the particle<\/td>kilogram (kg)<\/td><\/tr>\n
v<\/strong><\/td>Speed of the particle<\/td>metre per second (m\/s)<\/td><\/tr>\n
p<\/strong><\/td>Momentum, equal to m·v<\/td>kilogram metre per second (kg·m\/s)<\/td><\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n

Since 2019 the Planck constant has been fixed by definition rather than measured — its value is exactly 6.62607015 × 10-34<\/sup> J·s, as published in NIST’s CODATA tables<\/a>. The kilogram is now defined from it.<\/p>\n\n

Two rearrangements worth memorising<\/h3>\n\n

Exam questions rarely hand you the speed directly. Far more often you get a kinetic energy or an accelerating voltage, so these two forms save real time:<\/p>\n\n

\u03bb = h \/ sqrt(2\u00b7m\u00b7K)<\/div>\n\n
\u03bb = h \/ sqrt(2\u00b7m\u00b7q\u00b7V)<\/div>\n\n

Here K is kinetic energy in joules, q is the particle’s charge in coulombs, and V is the accelerating potential difference in volts. Both follow from p = sqrt(2mK), which is just kinetic energy<\/a> rearranged for momentum.<\/p>\n\n

If you only need the number and not the algebra, our De Broglie Wavelength Calculator<\/a> takes a mass and a speed and returns the wavelength with the working shown, which makes it a fast way to sanity-check your own answers.<\/p>\n\n

How the De Broglie Wavelength Works<\/h2>\n\n

De Broglie’s derivation starts with light and then makes one bold leap. Follow it in three steps.<\/p>\n\n

Step 1 — start with a photon.<\/strong> Einstein and Planck had already established that a light wave of frequency f carries energy in packets of E = hf, the basis of photon energy<\/a>.<\/p>\n\n

Step 2 — give the photon momentum.<\/strong> Relativity gives a massless particle p = E\/c. Substituting E = hf and c = fλ makes the frequency cancel, leaving p = h\/λ, or λ = h\/p.<\/p>\n\n

Step 3 — drop the assumption of masslessness.<\/strong> De Broglie’s leap was to claim λ = h\/p is not a fact about light at all. It is a fact about momentum, so it must apply to an electron exactly as it applies to a photon.<\/p>\n\n

Nothing in step 3 was justified when he wrote it. It was, in the most literal sense, a hypothesis — and three years later the electrons settled it.<\/p>\n\n

The experiment that proved it<\/h3>\n\n

In 1927, Clinton Davisson and Lester Germer at Bell Labs fired low-energy electrons at a nickel crystal and found scattering peaks exactly where wave diffraction<\/a> predicted them. Their 54 eV electrons behaved like a wave of roughly 0.165 nm.<\/p>\n\n

Plug 54 V into the formula and you get 0.167 nm. That agreement, from an experiment that began as an accident after a piece of glassware imploded, is documented in the American Physical Society’s account of the Davisson–Germer landmark<\/a>. George Paget Thomson found the same effect in thin films weeks later.<\/p>\n\n

\n \"Electron\n
An electron diffraction tube in operation. Electrons accelerated through a thin film strike the phosphor screen, the standard classroom demonstration of de Broglie’s prediction that matter diffracts.<\/figcaption>\n<\/figure>\n\n
<\/span>De Broglie Wavelength Lab<\/span><\/div>