Matter Wave and Uncertainty Principle

Matter Wave & Uncertainty

If light is both wave and particle, de Broglie asked whether particles are waves too. Smaller mass means longer wavelength and a more visible wave nature: electrons diffract, but a baseball's wavelength is far smaller than an atom. The uncertainty principle sets a limit on knowing position and momentum together. Slide momentum and kinetic energy here to see matter-wave wavelength and the uncertainty link.

Matter is also a Wave?

💡 De Broglie's Idea

①If light is both wave and particle, perhaps particles are also waves?

②De Broglie 1924: 'All matter has wave properties'

③Smaller mass → longer wavelength → wave nature observable

④Baseball: λ ≈ 10⁻³⁴ m — far smaller than an atom, never observable

⑤Electron: λ ≈ 10⁻¹⁰ m — comparable to atom size, diffraction observable!

De Broglie Wavelength and Momentum

Momentum p3

De Broglie Wavelength

λ = hp = hmv

wavelength = Planck constant / momentum

📐 Interpreting the Formula

①h = 6.63 × 10⁻³⁴ J·s (Planck constant — extremely small!)

②p = mv: larger mass → much smaller λ

③Larger momentum (speed) also reduces λ

④Wave nature is meaningful only for light particles like electrons

Energy–Wavelength Relation

Kinetic Energy E5

Energy–Wavelength

λ = h√(2mE)

Derived from KE = p²/2m

Photon Energy

E = hf = hcλ

Photon energy = Planck constant × frequency

🔬 Electron Diffraction Experiment

①1927 Davisson & Germer: shooting electrons at nickel produced diffraction patterns!

②Proved electrons have wavelengths similar to crystal lattice spacing

③First experimental confirmation of matter waves

④This is the principle of electron microscopes — using electron matter waves instead of light

Heisenberg's Uncertainty Principle

Uncertainty Principle

Δx · Δp ≥ h4π

position uncertainty × momentum uncertainty ≥ h/4π

🔍 Why Both Cannot Be Measured Precisely

①To see an electron, you must shoot light (photons)

②Short wavelength → precise position but high energy disturbs momentum

③Long wavelength → small momentum disturbance but uncertain position

④This is a fundamental law of nature, not a tech limitation!

⑤For macro objects, h is so tiny that uncertainty is negligible

Comparison

ChartClassical vs Quantum

Item	Classical Mechanics	Quantum Mechanics
Particle position	Precisely determined	Δx uncertainty exists
Momentum	Precisely determined	Δp uncertainty exists
Simultaneous measurement	Both can be measured precisely	Δx·Δp ≥ h/4π limit
Applies to	Macroscopic objects	Microscopic particles (electrons, protons)

Worked Examples

Example 1

If a particle’s momentum doubles, by what factor does its de Broglie wavelength change?

The de Broglie wavelength is λ = h/p, inversely proportional to momentum.

λ = hp

If p doubles, λ is halved.

p→2p ⇒ λ ∝ 12p = 12λ

▸ 1/2

The de Broglie wavelength is inversely proportional to momentum. Faster or heavier particles have shorter wavelengths, so their wave nature is hard to see.

Example 2

If a particle’s kinetic energy is increased fourfold, by what factor does its de Broglie wavelength change?

From λ = h/√(2mE), the wavelength is inversely proportional to √E.

λ = h√(2mE)

If E quadruples, √E doubles → λ is halved.

E→4E ⇒ λ ∝ 1√(4E) = 12λ

▸ 1/2

From λ = h/√(2mE), n× energy gives 1/√n × wavelength. 4× energy → √4 = 2, so half.

Summary

De Broglie Wavelength

λ = hmv

Wavelength of any matter = Planck constant / momentum

Uncertainty Principle

Δx · Δp ≥ h4π

Position and momentum cannot both be precisely measured

CSAT-style

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×

The uncertainty principle Δx·Δp ≥ h/4π sets a lower bound on the product.

Δx · Δp ≥ h4π

If Δx is halved, Δp must double to keep the bound.

Δx→12Δx ⇒ Δp ≥ 2 × h4πΔx

🎯 Exam Points

①De Broglie: λ = h/p = h/mv (mass ↑ → λ ↓)

②Electron diffraction = experimental evidence of matter waves

③Uncertainty: Δx·Δp ≥ h/4π (fundamental limit)

④Macro: h is so tiny that wave/uncertainty effects vanish

⑤Electron microscope: matter wavelength shorter than light → high resolution

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Interference & Diffraction

Special Relativity

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