Q.The mass of an electron is 9.1x10⁻³¹kg. If its kinetic energy is 3.0x10⁻²⁵J. Calculate its wavelength. (h=6.626x10⁻³⁴JS).
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🔒 Start your 14-day free trial to unlock the full solution →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.
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 wavelength λ is:
λ=ph=mvh
where:
- h=6.63×10−34 J⋅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.
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.
The Key Insight …
Using KE=½mv² to find velocity, then p=mv and λ=h/p, the electron's de Broglie wavelength works out to about 8.97×10⁻⁷ m (≈89 …
Using the kinetic energy to find momentum and then applying the de Broglie relation λ = h/p gives a wavelength of about 8.97×10⁻⁷ m.
Given: mass of electron m=9.1×10−31 kg, kinetic energy KE=3.0×10−25 J, h=6.626×10−34 J·s.
Step 1 — find momentum from kinetic energy:
KE=2mp2⟹p=2mKE
p=2×9.1×10−31×3.0×10−25=5.46×10−55
p=7.389×10−28 kg m s−1
…
- CBSE 2021Set annual1 markQ.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.
…
- CBSE 2019Set ANNUAL1 markMCQQ.The De-Broglie equation is:(a) λ = h/m(b) λ = h/c(c) λ = h/mc(d) λ = mc/h
›Reveal solutionSolution
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:
λ = h / p = h / (mv)
…
- CBSE 2018Set ANNUAL1 markMCQQ.De Broglie Equation is:(a) λ = h/m(b) λ = h/c(c) λ = h/mc(d) λ = hc/m
›Reveal solutionSolution
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).
Checking the printed options:
- (a) λ = h/m — dimensionally wrong, ignores velocity/momentum entirely.
- (b) λ = h/c — this is closer to the relation for photon momentum, not for a general moving particle. …
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