De Broglie Wavelength Formula: Pick the Right Form Fast

• The de Broglie wavelength is λ = h/p. Use h/(mv) when you have speed, h/√(2mK) when you have kinetic energy K, and h/√(2mqV) for a charged particle accelerated from rest through voltage V.
• For an electron, λ = 12.264/√V ångström with exact constants. Textbooks print 12.27 because of rounded constants, a gap of about 0.05%.
• By our calculation, the non-relativistic formula is within 1% for electrons below about 20 kV and about 4.8% off at 100 kV.
• Check units before you trust a result. An electron at tens to hundreds of volts has a wavelength of about one ångström (10⁻¹⁰ m). A result in seconds, or near 10⁻³⁴ m, means a step was missed.

The formula gives the wavelength λ that goes with a moving particle's momentum p. The relation is λ = h/p (OpenStax University Physics Vol. 3, Eq. 6.54). Everything else on this page is that one equation with p rewritten to suit the data you are given.

Momentum (p) is mass times velocity. Kinetic energy (K) is the energy of motion, K = p²/2m, so p = √(2mK). A charged particle (charge q) accelerated from rest through a voltage V gains K = qV. Substituting each into λ = h/p gives:

Rows 3 to 5 follow from the substitutions above, and the relativistic row is the same OpenStax momentum relation with E₀ = mc².

You need the particle's mass, plus one of speed, kinetic energy or voltage, and the constants below. Start from the electron at 100 V or at 2×10⁶ m/s, because both land in the range where the wavelength is about an ångström (Å, where 1 Å = 10⁻¹⁰ m).

The result comes out in metres. Divide by 10⁻⁹ for nanometres (nm) or by 10⁻¹⁰ for ångströms.

An electronvolt (eV) is the energy an electron gains through one volt, 1.602176634×10⁻¹⁹ J. If a question gives kinetic energy in eV, convert to joules before using h/√(2mK).

If you would rather run it on a laptop, this Python snippet uses the same formulas:

from math import sqrt
h = 6.62607015e-34; c = 299792458.0
e = 1.602176634e-19; m = 9.1093837139e-31   # electron

speed      = lambda m, v: h / (m * v)
voltage    = lambda m, q, V: h / sqrt(2 * m * q * V)
rel_volt   = lambda m, q, V: h * c / sqrt(q*V * (q*V + 2*m*c**2))

print(speed(m, 2e6))        # 3.637e-10 m
print(voltage(m, e, 100))   # 1.226e-10 m
print(rel_volt(m, e, 100e3))  # 3.701e-12 m

The shortcut is λ = h/√(2meV) with every constant folded into one number. Exact constants give h/√(2mₑe) = 12.264 Å·√V, so λ = 12.264/√V Å, or 1.2264/√V nm. For 100 V that is 12.264/10 = 1.226 Å, matching Example 2.

Printed values of 12.27 or 12.28 are consistent with rounded constants. Our recomputation (our model, not an official figure) shows how the choice of rounding moves the result:

The spread is under 0.2%. If a question hands you its own constants, use those. If it does not, use 12.27 and expect to land close to the printed answer.

The shortcut is for electrons only. For a proton or an α-particle, scale the electron result by √(mₑ/m) and divide by √(q/e).

The unit trap is the one that costs marks: 12.27/√V is in ångströms. For nanometres use 1.227/√V.

For electrons, the formula p = √(2mK) is good to within 1% up to about 20 kV and drifts past 5% above about 105 kV. It fails because it uses the classical kinetic energy, while the relativistic momentum is p = √[K(K + 2mc²)]/c (OpenStax).

We computed both versions for an electron with the exact constants listed earlier. Error is (λ non-relativistic ÷ λ relativistic) − 1.

The error reaches 1% at about 20.5 kV and 5% at about 105 kV. Algebraically, the relativistic form is equivalent to λ = 12.264/√[V(1 + eV/2mₑc²)] Å, which is the electron shortcut with a correction factor.

Our advice: if a question gives kilovolts, check the voltage before using 12.27/√V. At 10 kV or below the shortcut is safe. At 100 kV and above it is not.

Use the proportionality λ ∝ 1/√K for one particle. Because h and m stay fixed, constants cancel in a ratio, so you never need numbers. The same holds for voltage: λ ∝ 1/√V, since K = qV.

Check the wording. "Becomes 4 times" means a factor of 4. "Increased by 4 times" can be read as a factor of 5, so read the question carefully and state your reading.

For two different particles, compare using λ ∝ 1/√(mq) at the same voltage:

• Proton and α-particle at the same voltage: the α-particle has about 4 times the proton's mass and twice its charge, so mq is 8 times larger and λ(proton)/λ(α) = √8 ≈ 2.83.
• Same kinetic energy: λ ∝ 1/√m, so λ(proton)/λ(α) = √4 = 2.
• Same speed: λ ∝ 1/m, so the heavier particle has the shorter wavelength in proportion to its mass.

A question that changes kinetic energy or voltage and asks for a factor, or compares two particles under the same condition, is a ratio question. Practise that type with real past papers in our MHT-CET PYQ library.

Most wrong answers come from five slips, and each has a quick pass or fail test.

1. Kinetic energy used as momentum. Test: λ = h/K gives units of J s / J = seconds. Wavelength must be in metres, so this fails. Convert with p = √(2mK) first.
2. Mass without velocity. Test: h/m has units of m²/s. If your answer is not in metres, you dropped v.
3. Electronvolts not converted. Test: multiply by 1.602×10⁻¹⁹ to reach joules. An electron at 100 eV should give about 10⁻¹⁰ m. An answer of 10⁻⁸ m or 10⁻¹² m means the conversion is off.
4. Å versus nm. Test: 100 V should give 1.226 Å or 0.1226 nm. Reading "1.226" as nm is 10 times too large.
5. Wrong particle data. Test: for an α-particle use q = 2e and m about 4 times the proton's mass. Check that the wavelength drops as particles get heavier or more charged.

A quick dimension check catches 1 and 2 in seconds. Our units and dimensions page walks through that method, and how to solve physics numericals covers the habit of writing given data first.

We teach the full MHT-CET Physics syllabus in our MHTCET 2027 Long Term Batch, taught by Rahul Jewaani (RG Sir), who started RG Lectures in May 2018. De Broglie wavelength sits in the Dual Nature chapter of HSC Physics, and the batch covers the full syllabus with live classes, recordings, notes, previous-year questions (PYQs) and mock tests.

As of October 2026 the MHTCET 2027 Long Term Batch costs ₹1,632 and is valid until 31 May 2027 (batch page). It is listed on our store. For photon-energy numericals that sit next to this topic, see our photoelectric effect page. For the momentum-and-position side of the same ideas, see Heisenberg's uncertainty principle.

Padhna Aasaan Hai: pick the form, check the units, then do one line of arithmetic.