Wave and particle models of light

Historically, light was modelled as either a wave or a particle, but modern quantum physics recognises light can demonstrate both wave and particle characteristics.

De Broglie's hypothesis of matter waves

In 1924, Louis de Broglie proposed that just as light exhibits both wave and particle properties, particles should also demonstrate wavelike behaviour.

He related the wavelength ($\lambda$) to the momentum (p) of a particle:

$\lambda = \frac{h}{p}$

Where:

• $\lambda$ = Wavelength (m)
• h = Planck's constant (J s)
• p = momentum (kg m s^-1)

This is known as the de Broglie wavelength.

Initially met with skepticism, de Broglie's revolutionary idea was soon confirmed experimentally.

Confirming the wave properties of electrons

Electron diffraction experiments demonstrate the wave properties of electrons by showing diffraction patterns when electrons interact with crystals.

Key observations:

• Electron diffraction patterns have similar spacings to X-ray diffraction patterns
• Faster electrons diffract less than slower electrons

This evidence supports electron waves which have:

• Shorter wavelengths for faster moving electrons
• Longer wavelengths for slower electrons

These observations align with de Broglie's relationship between wavelength and momentum.

Relating de Broglie wavelength to momentum

The de Broglie equation can be rearranged to determine an electron's wavelength:

$\lambda = \frac{h}{mv}$

Where:

• $\lambda$ = Wavelength (m)
• h = Planck's constant (J s)
• m = Particle mass (kg)
• v = Particle speed (m s^-1)

Worked example - calculating an electron's de Broglie wavelength

An electron with mass $9.11 \times 10^{-31}$ kg moves at $7 \times 10^{6}$ m/s. Determine its de Broglie wavelength.

Step 1: Formula

$\lambda = \frac{\text{h}}{\text{m v}}$

Step 2: Substitution and correct evaluation

$\lambda = \frac{6.63 \times 10^{-34}}{(9.11 \times 10^{-31})(7 \times 10^{6})}=1.04\times10^{-10}\text{ m}$

Limits on observing quantum effects

Macroscopic particles do not demonstrate measurable wave effects because their de Broglie wavelengths are extremely small.

For diffraction to occur, the particle must interact with an object around the same size as its wavelength.

A 0.058 kg ball moving at 100 m/s has a wavelength of $1.14\times10^{-34}$ m, much smaller than atomic nuclei.