Wave-particle Duality

At the start of the 20th century, physicists assumed that electromagnetic radiation was exclusively a wave phenomenon. This model worked well for many phenomena — diffraction, interference, refraction — but it failed catastrophically when applied to the interaction between light and matter.

The solution came from a radical idea: electromagnetic radiation can also behave as discrete packets of energy. These packets are called photons. Each photon carries a fixed amount of energy that depends only on the frequency of the radiation:

Where:

• h = 6.63 × 10⁻³⁴ J s — Planck's constant
• f — frequency of the radiation (Hz)
• c = 3.00 × 10⁸ m s⁻¹ — speed of light
• λ — wavelength (m)

A photon is a quantum of electromagnetic radiation — the smallest possible unit of energy at a given frequency. You cannot have "half a photon" at a particular frequency; the energy comes in indivisible packets.

The Electronvolt

When working with photon energies, joules are often inconveniently small. The electronvolt (eV) is a more practical unit:

It is defined as the kinetic energy gained by an electron accelerated through a potential difference of 1 volt: E = eV = (1.60 × 10⁻¹⁹ C)(1 V) = 1.60 × 10⁻¹⁹ J.

Before the photon model, physicists assumed light was purely a wave. This worked for most phenomena, but when they tried to predict how much energy a hot object radiates (black-body radiation), classical physics gave a nonsensical answer: it predicted infinite energy at short wavelengths. This failure was called the ultraviolet catastrophe.

Planck's Solution: Energy Comes in Packets

In 1900, Max Planck fixed the problem with a radical idea. He proposed that the energy of the oscillators in a hot object is quantised — it can only take certain allowed values, not any value at all. These allowed energies are integer multiples of a fundamental unit:

Where h is Planck's constant (6.63 × 10⁻³⁴ J s) and n is an integer. This means energy is delivered in discrete packets of size hf — the same photons we discussed in Part 1.

The key insight: at high frequencies, the energy gap hf becomes so large that oscillators cannot easily jump to higher energy levels. This naturally prevents the energy from diverging to infinity — exactly what classical physics failed to predict.

Fig 1. Black-body radiation spectra at different temperatures (coloured curves) compared with the classical prediction (dashed). The classical theory wrongly predicts infinite energy at short wavelengths.

In 1924, a French physics student named Louis de Broglie proposed a bold idea that extended wave-particle duality beyond light. If waves (photons) can behave as particles, he argued, then perhaps particles can also behave as waves.

De Broglie suggested that any particle with momentum has an associated wavelength, given by:

Where:

• λ — de Broglie wavelength (m)
• h — Planck's constant (6.63 × 10⁻³⁴ J s)
• p — momentum of the particle (kg m s⁻¹)
• m — mass of the particle (kg)
• v — velocity of the particle (m s⁻¹)

What Does This Mean?

The de Broglie wavelength tells us the "size" of the wave associated with a particle. For everyday objects, λ is unimaginably small — far too small to detect. But for subatomic particles like electrons, λ can be comparable to the spacing between atoms in a crystal, making diffraction observable.

Relating λ to Kinetic Energy

For a particle accelerated from rest through a potential difference V, its kinetic energy is eV. Combining KE = ½mv² with p = mv and λ = h/p:

This form is extremely useful for electron diffraction calculations, where electrons are accelerated through a known potential difference.

De Broglie's hypothesis needed experimental confirmation. In 1927, Davisson and Germer demonstrated diffraction of electrons at Bell Labs. A simpler and more common school/college demonstration involves firing electrons through a thin film of polycrystalline graphite.

The Apparatus

1. Electron gun: A heated filament emits electrons, which are accelerated through a high potential difference (typically 2000–5000 V)
2. Graphite target: A very thin film of polycrystalline graphite. The carbon atoms are arranged in a hexagonal lattice with spacing of approximately d ≈ 10⁻¹⁰ m (about 0.1 nm)
3. Fluorescent screen: The diffracted electrons strike a phosphor screen, producing visible rings

What Is Observed

When the electron beam passes through the graphite film, a pattern of concentric rings is observed on the fluorescent screen. This is a diffraction pattern — clear evidence that electrons behave as waves.

Diffraction is a wave property. Particles, in the classical sense, cannot diffract. The fact that electrons produce a diffraction pattern proves they have an associated wavelength — the de Broglie wavelength.

Why Polycrystalline Graphite?

The graphite film consists of many tiny crystals oriented at random angles. Because the crystal orientations are random, the diffraction condition is satisfied for all angles simultaneously, producing concentric rings rather than discrete spots (which you would get with a single crystal).

How the Rings Relate to Atomic Spacing

The diameter of the diffraction rings on the screen depends on the de Broglie wavelength of the electrons and the spacing between the carbon atoms. For the first-order diffraction maximum, the condition is approximately:

Where d is the atomic spacing, θ is the diffraction angle, and n is the order of diffraction.

Effect of Changing the Accelerating Voltage

If the accelerating voltage is increased:

• Electrons gain more kinetic energy → higher momentum → shorter de Broglie wavelength
• The diffraction rings become smaller (closer together) because the wavelength is smaller relative to the atomic spacing

If the accelerating voltage is decreased:

• Electrons have less kinetic energy → lower momentum → longer de Broglie wavelength
• The diffraction rings become larger (further apart)

Wave-particle duality is the idea that all entities that carry energy and momentum (both matter and radiation) exhibit both wave-like and particle-like properties, depending on how they are observed.

The key insight is that both models are needed. Neither the wave model nor the particle model alone can explain all observations. The model you use depends on the experiment you are describing.

When Do We Use Each Model?

• Wave model for light: when studying propagation (diffraction, interference, refraction, polarisation)
• Particle model for light: when studying interactions with matter (photoelectric effect, emission/absorption spectra)
• Particle model for electrons: when studying motion in fields, electric current, collisions
• Wave model for electrons: when studying diffraction, interference, atomic orbitals, electron microscopy

Why Don't We Notice Wave Behaviour of Everyday Objects?

For macroscopic objects, the de Broglie wavelength is far too small to detect: