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De Broglie wavelength questions are short, but they hide several traps: which mass, which charge, kinetic energy or voltage, and when the simple formula stops working. This post builds every version from one equation, works through electrons, protons and alpha particles, and explains a misconception that research has measured.
What does the syllabus ask?
The JEE (Main) 2026 syllabus lists "matter waves: wave nature of particle, de-Broglie relation", and the NEET (UG) 2026 syllabus and CBSE 2025-26 curriculum use almost the same words. JEE Advanced 2026 lists "de Broglie wavelength of matter waves". The chemistry units of JEE Main also include de Broglie's relationship. Check the bulletin for your year.
Where does λ = h/p come from?
In 1924 Louis de Broglie proposed that moving particles have a wavelength λ = h/p, and he received the 1929 Nobel Prize in Physics "for his discovery of the wave nature of electrons". Clinton Davisson and Lester Germer reported electrons scattered by a single crystal of nickel in Nature in 1927, and Davisson shared the 1937 Nobel Prize with George Paget Thomson "for their experimental discovery of the diffraction of electrons by crystals". Our post on de Broglie and matter waves tells that story.
The NCERT chapter on dual nature gives λ = h/p = h/mv. Everything else follows by writing p in terms of what the question gives:
Speed given: p = mv.
Kinetic energy K given: K = p²/2m, so p = √(2mK) and λ = h/√(2mK).
Accelerating voltage V given, charge q: K = qV, so λ = h/√(2mqV).
Worked examples
Use h = 6.626 × 10⁻³⁴ J s, mₑ = 9.109 × 10⁻³¹ kg and e = 1.602 × 10⁻¹⁹ C, the CODATA values from NIST rounded to four figures.
Example 1: electron through 100 V.
p = √(2mₑeV) = √(2 × 9.109 × 10⁻³¹ × 1.602 × 10⁻¹⁹ × 100) = √(2.919 × 10⁻⁴⁷) = 5.40 × 10⁻²⁴ kg m/s.
λ = h/p = 6.626 × 10⁻³⁴ / 5.40 × 10⁻²⁴ = 1.23 × 10⁻¹⁰ m = 0.123 nm.
Putting the constants together once gives a shortcut for electrons: λ ≈ 1.227/√V nm, with V in volts. For V = 100, √V = 10 and λ = 0.1227 nm. Remember that this shortcut is only for electrons.
Example 2: proton through the same 100 V. Only the mass changes, and λ ∝ 1/√m. The proton is about 1836 times heavier, and √1836 ≈ 42.85, so λ = 0.1227/42.85 nm ≈ 2.86 × 10⁻¹² m, about 2.9 pm.
Example 3: proton and alpha particle through the same voltage. λ ∝ 1/√(mq). An alpha particle has about 4 times the mass and 2 times the charge, so λ_p/λ_α = √(4 × 2) = √8 ≈ 2.83.
Example 4: same kinetic energy. For equal K, λ ∝ 1/√m. For equal speed, λ ∝ 1/m. Write which quantity is equal before you write a ratio.
Example 5: electron or photon of the same energy. A 100 eV photon has λ = 1240/100 = 12.4 nm. A 100 eV electron has 0.123 nm, about a hundred times shorter. The photon formula λ = hc/E does not apply to particles with mass.
Why do students get this wrong?
Vokos, Shaffer, Ambrose and McDermott (2000) asked first-, second- and third-year physics students to predict how changing an experiment would change the pattern when electrons pass through slits or a crystal. Similar errors appeared at every level. One was a lack of working understanding of the de Broglie wavelength: students often treated it as a fixed property of the particle rather than a function of its momentum. If you speed an electron up, its wavelength gets shorter and its diffraction pattern narrows.
When does the simple formula stop working?
The formulas above use non-relativistic kinetic energy, K = p²/2m. That is fine while the kinetic energy is small compared with the rest energy. For an electron the rest energy is about 0.511 MeV, so voltages of a few kilovolts are safe, while at hundreds of kilovolts the error becomes noticeable. School and entrance questions almost always intend the simple form unless they say otherwise.
Common mistakes and how to avoid them
Using 1.227/√V for a proton or alpha particle. It is for electrons only.
Forgetting the charge. An alpha particle through V gains 2eV, not eV.
Squaring instead of rooting. λ ∝ 1/√V, so four times the voltage halves λ.
Mixing nm and m. Convert at the end, not halfway through.
Using λ = hc/E for an electron. That formula is for photons.
If the link between momentum and wavelength still feels like a formula rather than an idea, the Fermi Concept Library covers every concept in physics, each with its own lesson. For the light side of the chapter, see our post on Einstein's 1905 papers.
A practice routine for this chapter
Day 1: derive λ = h/√(2mqV) and the 1.227/√V shortcut from constants.
Day 2: six calculations: electrons at 25, 100 and 400 V, then a proton, a neutron and an alpha particle.
Day 3: six ratio questions, two each for equal voltage, equal kinetic energy and equal speed.
Day 4: three questions comparing photons and electrons of equal energy.
Day 5: a timed mixed set from previous papers.
Tonight, find the de Broglie wavelength of an electron accelerated through 400 V and check that it is half the 100 V value.
