Concept understanding — de Broglie Wavelength of Electron
The Intuition: Why Would an Electron Have a Wavelength?
Before 1924, physics had a clean split: light was a wave (interference, diffraction), and electrons were particles (they had mass, charge, followed Newton's laws). Then came experiments that blurred the line. Light, which everyone knew was a wave, also behaved like a stream of particles (photons) in the photoelectric effect. Einstein had shown that a photon's energy is E=hf, and its momentum is p=h/λ.
Louis de Broglie asked a bold, symmetrical question: if light — a wave — can behave like a particle, why can't a particle — like an electron — behave like a wave? Nature, he argued, might be symmetric. Every moving object should have a wavelength associated with it.
Note
This was pure theoretical insight in 1924. De Broglie had no experimental proof at the time — he was guided by the beauty of symmetry in physics. Three years later, Davisson and Germer proved him right by diffracting electrons off a nickel crystal.
The Precise Statement
For any particle with momentum p=mv, the associated de Broglie wavelengthλ is:
λ=ph=mvh
where:
h=6.63×10−34J⋅s (Planck's constant)
m is the mass of the particle (in kg)
v is its speed (in m/s)
For an electron, this is not a mathematical trick — it is a real, measurable wave property. The electron's wave nature is not visible in everyday life because the wavelength is incredibly small for macroscopic objects, but for an electron moving at typical speeds, it becomes comparable to atomic spacings in crystals.
λ=mvh
Why It Matters for an Electron
Take an electron accelerated through a potential difference V. Its kinetic energy comes from the electrical work done:
21mv2=eV
Solving for velocity and plugging into the de Broglie relation gives a very handy formula:
λ=2meVh
For V=100 volts, this works out to about 1.23×10−10 m — roughly the spacing between atoms in a crystal. That is why electrons can be diffracted by crystals, exactly like X-rays. The wave nature of the electron is not a philosophical curiosity; it is the operating principle of the electron microscope.
Watch out
A common mistake is to think the electron "turns into" a wave. It does not. The electron remains a single entity — it is both particle and wave simultaneously. The de Broglie wavelength describes the wave aspect of its behaviour, not a physical rippling in space.
a. Combining E=mc2 (mass-energy) with E=hν (photon energy) and ν=c/λ, and equating momentum p=mc (photon) to p=mv (particle), gives λ=h/mv.
b. Two limitations of Bohr's model: it fails for multi-electron atoms/spectra; it cannot explain the Zeeman/Stark splitting of spectral lines, and it conflicts with Heisenberg's uncertainty principle. …
(a) De Broglie's equation λ=h/mv is derived by combining E=mc2 and E=hν. (b) Bohr's model fails for multi-electron atoms and the Zeeman/Stark effects, and violates the uncertainty principle.
Part (a) — Derivation of de Broglie's equation:
For a photon, using Einstein's mass-energy relation, E=mc2.
Also, from Planck's quantum theory, E=hν=λhc.
Equating: mc2=λhc⇒mc=λh⇒λ=mch.
de Broglie extended this to any moving particle of mass m and velocity v (momentum p=mv), proposing that matter also has an associated wavelength:
λ=mvh=ph
Part (b) — Two limitations of Bohr's model:
It could explain the spectrum of hydrogen (a one-electron species) but failed to explain the spectra of multi-electron atoms. …
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
CBSE 2021Set annual1 mark
Q.Give answer of the following question in brief:
(i) What is de Broglie equation?
›Reveal solutionSolution
De Broglie equation: lambda = h/(mv), relating a particle's wavelength to its momentum.
Louis de Broglie proposed that matter, like light, has a dual (particle and wave) character. He suggested that a moving particle of mass m and velocity v has an associated wavelength lambda given by:
lambda = h / (m v) = h / p
where h is Planck's constant (6.626 x 10^-34 J s) and p = mv is the particle's momentum. This wave nature is significant (measurable) only for very small masses (like electrons) moving at appreciable speeds; for everyday macroscopic objects, the associated wavelength is far too small to observe.
The de Broglie relation is λ = h / (mv): wavelength equals Planck's constant divided by momentum (mass × velocity).
Louis de Broglie proposed that matter, like light, has a dual (particle-wave) character. For a particle of mass m moving with velocity v, the momentum is p = mv, and the associated wavelength is:
The de Broglie wavelength of a particle is λ = h/mv (Planck's constant divided by momentum); of the four printed options only (c) has the right structure (h divided by a mass×speed term).
Louis de Broglie proposed that every moving particle has an associated wavelength, given by
λ = h / p = h / (mv)
where h is Planck's constant, m is the mass of the particle, and v is its velocity. This says wavelength is inversely proportional to momentum (mv).