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Grade 11-12 (age 16-18)

high school physics Matter Wave and Uncertainty Principle

Matter Wave & Uncertainty

If light is a wave that also behaves like a particle, then perhaps a particle such as an electron can also behave like a wave; that was de Broglie's idea. The wavelength of a matter wave is λ = h/p, inversely proportional to momentum, so a light electron has a wavelength about the size of an atom and shows diffraction, while a baseball's wavelength is around 10⁻³⁴ m and never shows up. The limit that a particle's position and momentum cannot both be known as precisely as we like at the same time is Heisenberg's uncertainty principle. In the pictures, changing the momentum and the kinetic energy shows how much shorter the matter wavelength becomes.

Is Matter a Wave Too?

💡 De Broglie's Idea
①If light is a wave that also has particle properties, might particles also have wave properties?
②In 1924 de Broglie proposed that all matter has a wave nature
③The wavelength is inversely proportional to momentum, so the smaller the mass, the longer the wavelength and the easier the wave nature is to see
④A baseball's wavelength is about 10⁻³⁴ m, far smaller than an atom, so there is no way to observe it
⑤An electron's wavelength is about 10⁻¹⁰ m, close to the spacing between atoms, so diffraction is observed in crystals

De Broglie Wavelength and Momentum

3
De Broglie Wavelength
λ = hp = hmv
Wavelength = Planck constant ÷ momentum
📐 Reading the Formula
①The Planck constant h = 6.63 × 10⁻³⁴ J·s is a very small number
②Since p = mv, the larger the mass, the much shorter the wavelength
③Even for the same particle, going faster raises the momentum and shortens the wavelength; doubling p in the picture halves λ
④So the wave nature stands out clearly only for light particles such as electrons

Kinetic Energy and Wavelength

5
Kinetic Energy and Wavelength
λ = h√(2mE)
Obtained by putting p = √(2mE), from the kinetic energy E = p²/2m, into λ = h/p
Photon Energy
E = hf = hcλ
The photon relation used for the photoelectric effect; the wavelength of a matter wave is found from λ = h/p
🔬 The Electron Diffraction Experiment
①In 1927 Davisson and Germer fired electrons at a nickel crystal and obtained a diffraction pattern
②It confirmed that the electron's wavelength is similar to the spacing between atoms in the crystal
③This was the first experimental confirmation of matter waves
④An electron microscope uses the matter waves of electrons, with wavelengths much shorter than light, to resolve much smaller details

Heisenberg's Uncertainty Principle

Uncertainty Principle
Δx · Δp ≥ h4π
position uncertainty × momentum uncertainty ≥ h/4π
🔍 Why Can't Both Be Measured Exactly?
①To see where an electron is, light (photons) must be shone on it
②Short-wavelength light pins down the position, but the photon's large momentum jolts the electron's momentum a lot
③Long-wavelength light jolts the momentum less, but the position becomes blurred
④This is not a limit of measuring technology but a limit built into nature
⑤Because h is so small, this uncertainty does not show up for everyday objects
⑥The narrower Δx is made, the larger the minimum Δp becomes, and the reverse also holds

Comparison

Classical Mechanics and Quantum Mechanics
ItemClassical MechanicsQuantum Mechanics
Particle positionCan be fixed exactlyHas an uncertainty Δx
MomentumCan be fixed exactlyHas an uncertainty Δp
Simultaneous measurementBoth values can be measured preciselyLimited by Δx·Δp ≥ h/4π
Applies toEveryday objectsMicroscopic particles such as electrons and protons

Worked Examples

Example 1
If a particle’s momentum doubles, by what factor does its de Broglie wavelength change?
1
The de Broglie wavelength λ = h/p is inversely proportional to momentum.
λ = hp
2
If p becomes 2 times larger, λ becomes 1/2 as large.
p→2p ⇒ λ ∝ 12p = 12λ
▸ 1/2
The faster or heavier a particle, the larger its momentum and the shorter its wavelength, so its wave nature is that much harder to see.
Example 2
If a particle's kinetic energy becomes 4 times larger, by what factor does its de Broglie wavelength change?
1
In λ = h/√(2mE), the wavelength is inversely proportional to √E.
λ = h√(2mE)
2
If E becomes 4 times larger, √E becomes 2 times larger, so λ becomes 1/2 as large.
E→4E ⇒ λ ∝ 1√(4E) = 12λ
▸ 1/2
When the energy becomes n times larger, the wavelength becomes 1/√n as large. Note that making the energy 4 times larger halves the wavelength rather than making it 1/4.

Summary

De Broglie Wavelength
λ = hmv
The wavelength of any matter = Planck constant ÷ momentum
Uncertainty Principle
Δx · Δp ≥ h4π
Position and momentum cannot both be measured as precisely as we like at the same time
CSAT-style problem, adapted
If an electron’s position is measured more precisely so the position uncertainty Δx is halved, how does the minimum momentum uncertainty Δp change?
①12×
②No change
③2×
④4×
⑤14×
▸ ③ 2×
1
In the uncertainty principle Δx·Δp ≥ h/4π, the product of the two uncertainties has a lower limit.
Δx · Δp ≥ h4π
2
If Δx becomes 1/2 as large, the minimum Δp must become 2 times larger to respect that limit.
Δx→12Δx ⇒ Δp ≥ 2 × h4πΔx
🎯 Exam Points
①The de Broglie wavelength λ = h/p = h/mv is shorter for larger mass or speed
②The electron diffraction experiment is the experimental evidence for matter waves
③The uncertainty principle Δx·Δp ≥ h/4π is a fundamental limit of nature
④For everyday objects h is so small that both the wave nature and the uncertainty can be ignored
⑤An electron microscope has high resolution because the electron's matter wavelength is shorter than that of light
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Interference & Diffraction of Light
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Special Relativity

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