9478 / 2027
Quantum Physics overview

Lesson 5 of 7

Localisation and momentum spread

  • I can explain localisation through a necessary momentum spread and apply the position-momentum width relation.Syllabus

    Syllabus 9478, 19(h). Explain localisation through a necessary momentum spread and apply the position-momentum width relation.

Make a guess. You are not marked.

A beam of electrons passes through a slit, which is then made narrower. What happens to the angular spread of the beam beyond the slit?

  1. It narrows, because the slit now selects electrons moving straight ahead.
  2. It widens, because a narrower position range means a larger momentum spread.
  3. It disappears, because electrons cannot pass a gap narrower than themselves.
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It widens, because a narrower position range means a larger momentum spread.

The slit limits where each electron can be across the beam. A narrower slit means a smaller spread in position, so a larger spread in sideways momentum, and the beam fans out more.

A sharply localised state needs a spread of momentum components. This is a property of the state, not merely poor measuring technique or an uncertainty about one definite classical speed.

A single perfectly defined travelling-wave momentum does not form a localised packet. Combining momentum components can localise the state, with a narrower position distribution requiring a broader momentum distribution along the same axis.

Δx Δp ≳ h

Here Δx and Δp are characteristic width scales, and the relation means greater than or of the order of h. Use it for an order-of-magnitude estimate. It is not an exact equality and does not require a state to attain a minimum product. It is also not an instrument specification.

Compare position widths on one common scale

Compare position widths on one common scaleTwo normalized Gaussian position densities have the same mean of zero. The horizontal axis is X equals position divided by reference width a, from minus four to plus four. The vertical axis is a times position density, so both axes are dimensionless. Solid state A has unit width and peak 0.398942 on these relative axes. Dashed state B has width 0.5 and peak 0.797885. Both curves use exactly the same horizontal and vertical scales within the panel. Their full tails continue beyond the displayed window and their full areas are one. The corresponding states use widths a and a over two in position, and b and two b in momentum.00.20.40.60.8-4-2024a × position density(dimensionless)Scaled position X = x/aState AState B

State B has half the position width of A and twice its peak density. Both mean positions are zero. These relative Gaussian shapes illustrate corresponding spreads; they do not replace the separate syllabus order-of-magnitude estimate.

The narrower-position state has a broader momentum distribution

The narrower-position state has a broader momentum distributionTwo normalized Gaussian momentum densities have the same mean of zero. The horizontal axis is P equals momentum divided by reference width b, from minus six to plus six. The vertical axis is b times momentum density, so both axes are dimensionless. Solid state A has unit width and peak 0.398942 on these relative axes. Dashed state B has width 2 and peak 0.199471. Both curves use exactly the same horizontal and vertical scales within the panel. Their full tails continue beyond the displayed window and their full areas are one. The corresponding states use widths a and a over two in position, and b and two b in momentum.00.20.40.60.8-6-3036b × momentum density(dimensionless)Scaled momentum P = p/bState AState B

Follow the same state labels: B is twice as broad in momentum, although both mean momenta remain zero. Its peak is half as high so the full density area remains 1. The tails continue beyond this crop.

Both states have the same mean position and the same mean momentum. On a common scale, halving the position width accompanies doubling the momentum width. Each complete density has area 1 and continuing tails; the vertical axes are scaled densities, not probabilities at a single point.

The momentum spread describes the distribution of possible results for similarly prepared states. Its mean can remain zero while its width is large. Reducing ruler or detector errors does not remove this state-level relationship.

Worked width-scale estimate

Estimate momentum spread before a velocity scale

Take an electron localised on the scale Δx = 2.0 × 10-10 m. With h = 6.63 × 10-34 J s:

Characteristic required Δp is of order h/Δx
= (6.63 × 10-34)/(2.0 × 10-10)
= 3.315 × 10-24 kg m/s
≈ 3.3 × 10-24 kg m/s

Using the non-relativistic electron mass me = 9.11 × 10-31 kg gives a corresponding velocity-spread scale:

Δv of order Δp/me
= 3.64 × 106 m/s

This is a spread scale, not a measured speed or a required mean velocity. Halving Δx to 1.0 × 10-10 m doubles the characteristic required momentum and velocity spreads under the same convention. It does not require the mean momentum to double.

Why the width convention matters

The estimate above uses characteristic widths and suppresses numerical factors. If position and momentum widths are defined specifically as standard deviations σx and σp, their precise lower bound is σxσp ≥ h/(4π). Do not substitute standard-deviation numbers into the rough-width form while treating it as an exact theorem.

Check your understanding

Using the same order-of-magnitude position-momentum width convention, a particle is localised to half its previous position-width scale. How does the characteristic required momentum-spread scale change?

  1. It doubles; the mean momentum need not double.
  2. It halves, because the particle has less space to move.
  3. The mean momentum must double and the spread stays unchanged.
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It doubles; the mean momentum need not double.

The required spread scale is of order h/Delta x. Halving the position scale doubles that momentum-spread scale. A distribution can retain zero mean while becoming wider.