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de Broglie Hypothesis: Wave Nature of Matter

The de Broglie hypothesis states that every moving material particle is associated with a matter wave whose wavelength is λ = h/p, where p is the momentum of the particle.

In one line: Matter wavelength becomes visible when momentum becomes tiny.

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Understand it

  • Light shows both wave and particle behaviour, and de Broglie proposed that moving particles should also show wave behaviour.
  • For a non-relativistic particle of mass m moving with speed v, p = mv, so λ = h/mv.
  • The wavelength is noticeable for microscopic particles such as electrons but extremely small for everyday objects because their momentum is large.
  • Electron diffraction provides evidence for the wave nature of matter.

Example

An electron accelerated through a potential difference has a measurable de Broglie wavelength, which allows electron beams to produce diffraction patterns from crystals.

Formula and rule

de Broglie wavelength: λ = h/p.

For a non-relativistic particle, λ = h/mv.

For a charged particle accelerated through potential V, kinetic energy qV = p2/(2m), so λ = h/√2mqV.

For an electron, q = e.

SI units: λ in m, h in J s, p in kg m s−1, m in kg, v in m s−1, q in C, and V in volt.

Use only when the particle is moving and non-relativistic conditions are assumed for mv form.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

An electron is accelerated through 100 V.

  1. 1Using λ = h/√2meV, λ = (6.63 × 10−34)/√(2 × 9.11 × 10−31 × 1.6 × 10−19 × 100).
  2. 2The denominator is about 5.40 × 10−24 kg m s−1, so λ = 1.23 × 10−10 m.
  3. 3This wavelength is of atomic size, so electron diffraction is possible.

Method you can reuse

Identify the moving particle, choose p = mv or momentum from acceleration voltage, substitute SI values, calculate λ, and compare the result with atomic dimensions if the question asks for physical significance.

Visual tip

For Davisson-Germer evidence, a diagram may show an electron beam incident on a nickel crystal, scattered electrons, detector, and intensity variation with angle. For formula use, a wave drawn along the particle path can support the idea.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is de Broglie wavelength not observed for a moving cricket ball in ordinary conditions?

Think first, then show answer
A cricket ball has very large momentum compared with an electron, so λ = h/p gives an extremely small wavelength. The wave behaviour is therefore not observable in ordinary situations, while it becomes important for microscopic particles.

Common wrong answer

Wrong: A particle at rest has a de Broglie wavelength h/m. Correct: de Broglie wavelength is associated with momentum; if p = 0, λ = h/p is not meaningful as a finite matter wavelength for motion.

Don't mix this up

Students often apply λ = h/mv to photons. For photons, use p = h/λ or E/c; λ = h/p is general, but mv is not used for photons because photons have no rest mass.

Exam tip

For electron acceleration numericals, first find momentum from kinetic energy or use the standard electron relation if allowed. Always keep mass in kg, charge in coulomb, potential in volt, and wavelength in metre.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Asked as a statement of hypothesis, derivation of λ = h/√2meV, numerical wavelength of an electron, comparison with macroscopic objects, and significance of Davisson-Germer experiment.
  1. 1

    Enough for 1 mark

    The de Broglie hypothesis states that every moving material particle is associated with a matter wave whose wavelength is λ = h/p, where p is the momentum of the particle.

  2. 2

    Add for 2 marks

    Example: An electron accelerated through a potential difference has a measurable de Broglie wavelength, which allows electron beams to produce diffraction patterns from crystals.

  3. 3

    Add for 3 marks

    Light shows both wave and particle behaviour, and de Broglie proposed that moving particles should also show wave behaviour.

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