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Heisenberg's Uncertainty Principle (A Level)

Key idea: Use Heisenberg’s uncertainty principle to link position localisation to momentum spread, and solve ΔxΔp questions with exam-safe wording (A Level Physics).

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Interpret wavefunctions, probability density and superposition.
  • Apply uncertainty and infinite-square-well energy quantisation.

1. Definitions (Must Know)

A. Heisenberg uncertainty principle (position–momentum)

There is a fundamental limit to how precisely position and momentum can be known simultaneously:

Δ x Δ p ≳ h

A more precise common form is:

Δ x Δ p ≥ ħ/2

where ħ = h/2π.

B. Meaning of Δ x and Δ p

  • Δ x is the uncertainty (spread) in position.
  • Δ p is the uncertainty (spread) in momentum.

These are not “instrument errors”; they are linked to the wave nature of particles.

2. Key Ideas (What Earns Marks)

  • More localisation (smaller Δ x) implies larger momentum spread (larger Δ p).
  • The uncertainty principle is about the state and the wavefunction spread, not just “disturbance by measurement”.
  • A useful link to de Broglie: p = h/λ A short wavelength (good spatial resolution) corresponds to large momentum.
Safe wording

Say “localising a particle implies a spread of momenta” rather than “the instrument is inaccurate”.

3. Detailed Explanations

A. Why wave nature implies uncertainty

To localise a particle, its wavefunction must be “narrow” in space.

A narrow wave packet requires a superposition of many wavelengths, meaning many momenta (since p = h/λ). So the momentum is uncertain.

B. Measurement “disturbance” is not the main point

It is true that measurements can disturb microscopic systems, but even with an ideal, non-disturbing measurement, quantum states have intrinsic spreads set by the uncertainty principle.

4. Common Mistakes

  • Treating the uncertainty principle as a limitation of instruments.
  • Using h and ħ interchangeably without the factor 2π.
  • Mixing up momentum uncertainty Δ p with speed uncertainty Δ v (you can relate them using p = mv for non-relativistic motion).

5. Exam Tips

  • If a question gives Δ x, you can estimate: Δ p ≈ ħ/(2Δ x)
  • If you need speed spread (non-relativistic): Δ p = mΔ v
  • Always state if you are using the “ħ/2” form or an order-of-magnitude “∼ h” estimate.

6. Worked Examples

Modelled example 1

Estimate momentum uncertainty from position uncertainty

Core

Problem

An electron is localised to within Δ x = 1.0 × 10⁻¹⁰ m. Using Δ x Δ p ≥ ħ/2, estimate the minimum Δ p. Take ħ = 1.05 × 10⁻³⁴ J s.
Study the worked solution

  1. Isolate momentum spread

    Method

    Δ p ≥ ħ/(2Δ x).

    Reason

    The minimum estimate corresponds to the lower bound of the supplied uncertainty relation.

    Working

    Δ p ≥ (1.05 × 10⁻³⁴)/(2(1.0 × 10⁻¹⁰))

  2. Evaluate

    Method

    Δ pₘᵢₙ = 5.3 × 10⁻²⁵ kg m s⁻¹.

    Reason

    Stronger localisation would require a larger minimum momentum spread.

    Working

    Δ pₘᵢₙ = 5.3 × 10⁻²⁵ kg m s⁻¹

Common misconception 2

Estimate speed uncertainty

Find and correct the mistake

Learner claim

Using Δ p = 5.3 × 10⁻²⁵ kg m s⁻¹ from Example A, a learner writes Δ v = mₑΔ p. Diagnose the rearrangement and estimate the electron’s minimum speed uncertainty. Take mₑ = 9.11 × 10⁻³¹ kg and assume non-relativistic motion.

Try this before viewing the solution

Unit: m s^-1

View solution step by step

  1. Repair the rearrangement

    Method

    Δ v = Δ p/mₑ.

    Reason

    For fixed non-relativistic mass, Δ p = mₑΔ v; multiplying again by mass is dimensionally wrong.

    Working

    Δ v = (5.3 × 10⁻²⁵)/(9.11 × 10⁻³¹)

  2. Evaluate

    Method

    Δ v = 5.8 × 10⁵ m s⁻¹.

    Reason

    A very small electron mass converts the momentum spread into a substantial speed spread.

    Working

    Δ v = 5.8 × 10⁵ m s⁻¹

7. Mind Stretchers

Mind stretcher 1: Bigger box, smaller momentum spreadExtension

Explain why a particle in a larger 1D box tends to have a smaller momentum uncertainty.

Show Answer

If the particle is confined to a region of width Δ x∼ L, then the uncertainty principle gives: Δ p ≳ ħ/2L

So increasing L decreases the minimum momentum spread.

8. Optional (Enrichment)

A. Energy-time uncertainty (not required here)

Some texts discuss an energy–time uncertainty relation. In this syllabus track, the high-frequency assessed form is the position–momentum relationship above.

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027