Explain how the de Broglie hypothesis accounts for the discrete energy levels observed in the hydrogen atom.

IB DP Physics Higher Level (2023 syllabus) — E.2 Quantum physics (HL only) · Explain · 4 marks · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

Niels Bohr's model of the hydrogen atom postulated that electrons can only occupy certain allowed orbits without radiating energy. Louis de Broglie later proposed that particles such as electrons possess wave-like properties characterised by a wavelength λ = h/p.

Model answer (4 marks)

The electron is associated with a de Broglie wavelength λ = h/p (or λ = h/mv). Only orbits in which an integer number of de Broglie wavelengths fit exactly around the circumference are permitted – the standing‑wave condition nλ = 2πr. If the standing‑wave condition is not met, destructive interference causes the wave to cancel, so that orbit is not stable and is not permitted. Because only certain radii are allowed by the standing‑wave condition, only certain momenta (and hence kinetic energies) are permitted, giving rise to discrete energy levels.

Examiner tips

  • Mention the standing‑wave condition explicitly; link it to allowed radii.
  • Explain why non‑integer wavelengths lead to destructive interference.
  • Show that allowed radii give discrete momenta and energies.
  • Use the correct symbols (h, p, m, v, n).

Common mistakes

  • Forgetting to state the standing‑wave condition nλ = 2πr.
  • Confusing the de Broglie wavelength with the orbit circumference without the integer factor.
  • Using the wrong expression for the wavelength (e.g., λ = h/mv without noting p = mv).

Mark scheme (4 marks)

  1. The electron is associated with a de Broglie wavelength given by λ = h/p (or λ = h/mv).
  2. Only orbits in which an integer number of de Broglie wavelengths fit exactly around the circumference are permitted (standing wave condition: nλ = 2πr).
  3. If the standing wave condition is not met, destructive interference causes the wave to cancel, so that orbit is not stable/not permitted.
  4. Because only certain radii are allowed by the standing wave condition, only certain momenta (and hence kinetic energies) are permitted, giving rise to discrete energy levels.

Key terms in this question

discrete energy levels

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