Electron Wavelength Calculator and de Broglie Formula

An electron wavelength calculator uses an electron’s kinetic energy or momentum to find its de Broglie wavelength. Enter kinetic energy in electronvolts and the calculator returns the wavelength in picometers, along with the corresponding energy in joules.

Formula

The de Broglie relation connects a particle’s wavelength to its momentum:

λ = h / p

For an electron with kinetic energy K, the relativistic energy–momentum relation gives:

λ = hc / √[K(K + 2m_ec²)]

Here, λ is wavelength in metres, h is Planck’s constant, p is momentum in kilogram-metres per second, K is kinetic energy in joules, m_e is the electron rest mass in kilograms, and c is the speed of light in metres per second. The calculator accepts E, the numerical value of kinetic energy in electronvolts. It uses the equivalent convenient form λ (pm) = 1226.426 / √[E(1 + E/1,022,000)]. The constants are expressed so that the answer is in picometers.

The first output is the wavelength, rounded to three decimal places. The second converts the entered energy to joules using 1 eV = 1.602176634 × 10⁻¹⁹ J. The relativistic correction is included; for energies far below an electron’s rest energy, it makes only a small difference.

Worked Example

Using the default kinetic energy, E = 100 eV:

1. Insert the energy into the wavelength formula: λ = 1226.426 / √[100(1 + 100/1,022,000)] pm.
2. Evaluate the term in brackets: 100(1 + 100/1,022,000) ≈ 100.009784.
3. Take its square root and divide: λ ≈ 1226.426 / 10.000489 = 122.636 pm.
4. Convert the energy to joules: 100 × 1.602176634 × 10⁻¹⁹ = 1.602177 × 10⁻¹⁷ J.

The calculator therefore reports an electron wavelength of 122.636 pm and kinetic energy of 1.602177 × 10⁻¹⁷ J.

What the Wavelength Means

The de Broglie wavelength is the wavelength associated with a particle’s momentum. It does not mean that an electron is a tiny ball physically oscillating with a visible surface wave. Quantum mechanics describes the electron with a wavefunction; the de Broglie wavelength helps characterize how that wave behaves in situations such as diffraction and interference.

Wavelength is inversely proportional to momentum: increasing an electron’s kinetic energy increases its momentum and decreases its wavelength. At 100 eV, the calculated wavelength is about 0.123 nm. This scale is comparable to distances between atoms in solids, which is why electron diffraction can reveal information about crystal structure. Whether a particular experiment shows clear diffraction also depends on the electron source, sample, geometry, and interactions—not wavelength alone.

When to Use an Electron Wavelength Calculator

Use this calculation when you know an electron’s kinetic energy and want its associated wavelength, for example when estimating a diffraction scale or checking a physics problem. The result assumes that the entered energy is the electron’s kinetic energy. It does not calculate the wavelength of a photon, a matter wave for a different particle, or an electron’s full quantum state.

• Energy in eV: A common unit for particle energies. One electronvolt is the energy gained by a particle with one elementary charge moving through a potential difference of one volt, in the idealized energy conversion.
• Wavelength in pm: One picometer is 10⁻¹² metres. To convert picometers to nanometers, divide by 1,000; to convert to ångströms, divide by 100.
• Energy in joules: The calculator supplies this conversion for use in SI equations. It is not a second measurement or an independent input.

Relativistic and Nonrelativistic Results

At low kinetic energies compared with the electron rest energy, approximately 511,000 eV, the classical momentum-energy relation is a useful approximation. It leads to λ ≈ h/√(2m_eK). With K in electronvolts, this is approximately λ (pm) = 1226.4/√E. The calculator’s full expression adds the relativistic correction, so it is appropriate across a broader energy range.

For example, at 100 eV the simple approximation gives about 122.64 pm, very close to the relativistic result of 122.636 pm. As energy rises, the difference grows. At energies approaching or exceeding the electron rest energy, use a relativistic formula rather than treating the classical approximation as exact.

Common Mistakes and Limits

• Entering voltage as energy without checking the setup: An electron accelerated through a potential difference of V volts gains kinetic energy of V electronvolts only under the usual assumption that it starts with negligible kinetic energy and is accelerated through that full potential difference. If it has initial energy or loses energy, account for that separately.
• Mixing units: The input is eV, not joules, keV, or volts. Convert first: 1 keV = 1,000 eV. The output wavelength is in pm, not nm or metres.
• Confusing wavelength with resolution: A shorter wavelength can support finer-scale probing, but it does not guarantee that an instrument will resolve that scale. Experimental conditions and electron interactions matter.
• Overinterpreting rounded digits: Three decimal places are a display choice, not a statement that the input energy is known to that precision. Report a result consistent with the precision of the measured or supplied energy.

The calculation gives the free-particle de Broglie wavelength based on kinetic energy. It does not include material-specific effects such as an electron’s interactions inside a crystal, nor does it determine whether a beam is coherent or monochromatic. For a different particle, substitute that particle’s mass and charge-independent momentum relation rather than using this electron-specific calculator.

Frequently Asked Questions

What is the formula for an electron’s wavelength?
The basic relation is λ = h/p, where h is Planck’s constant and p is momentum. If kinetic energy is known, the relativistic form is λ = hc/√[K(K + 2m_ec²)].

How do I calculate an electron wavelength from volts?
If an electron starts nearly at rest and gains energy across a potential difference V, use kinetic energy K = V eV, then enter that energy in the calculator. Otherwise, determine the electron’s actual kinetic energy first.

Does higher electron energy mean a shorter wavelength?
Yes. Higher kinetic energy means greater momentum, and λ = h/p, so the associated wavelength becomes shorter.

Why does the calculator use a relativistic formula?
The relativistic expression remains valid when the electron’s speed is high. At ordinary low energies it gives nearly the same result as the nonrelativistic approximation, while avoiding that approximation’s growing error at higher energies.

What is the wavelength of a 100 eV electron?
Using the relativistic de Broglie relation, a 100 eV electron has a wavelength of about 122.636 pm, or 0.122636 nm.

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in E | Kinetic energy | eV | 100 | 0
out lambda | Electron wavelength | pm | 1.226426 * 1000 / sqrt(E * (1 + E / 1022000)) | 3
out joules | Kinetic energy | J | E * 1.602176634e-19 | 6