Explain how Heisenberg's uncertainty principle accounts for the fact that electrons cannot exist inside an atomic nucleus.

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).

The diameter of a typical atomic nucleus is of the order 10⁻¹⁴ m. An electron has a rest mass of 9.11 × 10⁻³¹ kg.

Model answer (4 marks)

If an electron were confined to the nucleus, its positional uncertainty would be Δx≈10⁻¹⁴ m. By Heisenberg’s principle ΔxΔp≥ħ/2, so Δp≥ħ/(2Δx)≈1.05×10⁻³⁴ J·s/(2×10⁻¹⁴ m)≈2.6×10⁻²¹ kg·m s⁻¹. This gives a minimum momentum p_min≈Δp and a corresponding kinetic energy E_k≈p_min²/(2m_e)≈(2.6×10⁻²¹)²/(2×9.11×10⁻³¹)≈3.7×10⁻¹⁰ J≈2.3 MeV, far larger than the electron’s rest‑mass energy (0.511 MeV). Thus the energy required to confine the electron inside the nucleus is prohibitively high, so the uncertainty principle prevents electron confinement inside an atomic nucleus.

Examiner tips

  • State Δx≈10⁻¹⁴ m and use ΔxΔp≥ħ/2 to find Δp
  • Calculate the minimum kinetic energy and compare with rest‑mass energy
  • Explain why this makes confinement impossible
  • Use correct units and symbols

Common mistakes

  • Using Δx≈10⁻¹⁵ m instead of 10⁻¹⁴ m
  • Confusing ħ with h or missing the 1/2 factor
  • Failing to convert kinetic energy to MeV or compare with rest‑mass energy

Mark scheme (4 marks)

  1. If an electron were confined within the nucleus, its positional uncertainty Δx would be of the order of the nuclear diameter (~10⁻¹⁴ m).
  2. By the uncertainty principle ΔxΔp ≥ ℏ/2 (or h/4π), a small Δx requires a correspondingly large minimum uncertainty in momentum Δp.
  3. The large momentum uncertainty implies a minimum momentum (and hence minimum kinetic energy) far exceeding the rest-mass energy of the electron, making confinement energetically impossible.
  4. Therefore the uncertainty principle rules out electron confinement inside a nucleus, as no physical mechanism can supply the necessary energy to maintain such confinement.

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