de Broglie’s Matter Waves (HSC SSCE Physics): Revision Notes

de Broglie's Matter Waves

Introduction to wave-particle duality

For centuries, scientists understood light exclusively as a wave phenomenon. Light displays classic wave behaviours including reflection, refraction, diffraction and interference. However, in 1905, Einstein's explanation of the photoelectric effect required a radical shift in thinking. To explain his observations, Einstein proposed that light behaves as discrete particles (photons), where each photon carries energy equal to Planck's constant multiplied by its frequency: $E = hf$.

This discovery led to the concept of wave-particle duality - the idea that light simultaneously exhibits both wave and particle characteristics. The key to understanding this apparent contradiction lies in recognising that the wave model and particle model complement each other. When examining properties like diffraction and interference, the wave model applies. When explaining phenomena such as the photoelectric effect, the particle model proves more useful.

Deriving the relationship between waves and particles

To describe wave-particle duality quantitatively, physicists combined two fundamental equations. Einstein's mass-energy equivalence states:

$E = mc^2$

Planck's hypothesis for photon energy states:

$E = hf$

Setting these equal gives:

$mc^2 = hf$

Since the frequency $f$ and wavelength $\lambda$ of light are related by $c = f\lambda$, we can substitute $f = \frac{c}{\lambda}$:

$mc^2 = \frac{hc}{\lambda}$

Rearranging for wavelength:

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

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

where:

• $\lambda$ is the wavelength of light (measured in metres)
• $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s)
• $m$ is the mass of the photon (in kg)
• $c$ is the speed of light ($3.00 \times 10^8$ m·s$^{-1}$)
• $mc$ represents the momentum of the photon (in kg·m·s$^{-1}$)

de Broglie's matter waves hypothesis

In 1924, French physicist Louis de Broglie made a bold theoretical leap. He proposed that wave-particle duality applies not just to light, but to all matter.

de Broglie generalised the wave-particle equation by replacing $c$ (the speed of light) with $v$ (the velocity of any particle):

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

This equation suggests that any particle possessing momentum exhibits wave-like properties. The wavelength associated with a moving particle is called its de Broglie wavelength or matter wave. This concept proves particularly significant for small particles like electrons, as demonstrated in the following calculations.

Calculating de Broglie wavelengths

For everyday objects, the momentum $p = mv$ determines the de Broglie wavelength. Let's examine what this means through a worked example.

This wavelength is extraordinarily small - far too tiny to observe. Although all moving objects technically possess wave characteristics, the wavelengths of massive objects are imperceptibly small. This explains why we don't observe wave behaviour in everyday objects like tennis balls or cars.

Experimental evidence for matter waves

de Broglie's theoretical proposal required experimental verification. The crucial evidence came from electron diffraction experiments.

Understanding diffraction and interference

To understand these experiments, we must first review two wave properties:

Diffraction is the bending of waves as they pass around obstacles or through narrow openings. When waves encounter a barrier with a slit comparable to their wavelength, they spread out in a characteristic pattern.

When waves pass through two adjacent slits, the diffracted waves from each slit overlap and interact, causing interference:

• Constructive interference occurs when wave crests align with crests (or troughs with troughs), producing larger amplitudes
• Destructive interference occurs when crests align with troughs, cancelling each other out

This creates an alternating pattern of maximum and minimum intensities. For visible light, this appears as bright and dark bands. For other types of waves, detectors measure varying signal intensities.

The Davisson-Germer experiment (1927)

Clinton Davisson and Lester Germer designed an experiment to test whether electrons exhibit wave properties.

Experimental setup:

• Electrons were accelerated to high velocity using a potential difference of 54 V
• The electron beam was directed at a heated nickel crystal
• A movable detector measured the intensity of scattered electrons at different angles

Key observations:

When electrons struck the nickel crystal surface, they scattered off different atomic planes. Some electrons passed through gaps between nickel atoms, which acted like multiple slits. If electrons behaved as waves, they should undergo diffraction and produce interference patterns.

Results:

The detector measured a series of intensity maxima and minima - exactly the pattern expected from wave interference. This proved that electrons possess wave properties. Furthermore, the measured wavelength from the interference pattern matched the value calculated using de Broglie's equation $\lambda = \frac{h}{mv}$.

GP Thomson's experiment (1928)

George Paget Thomson (son of JJ Thomson, who discovered the electron) provided additional evidence for matter waves.

Thomson directed an electron beam through a thin gold foil. The scattered electrons created a pattern on photographic film positioned behind the foil. He then compared this pattern with one produced by X-rays (which were known to exhibit wave properties) scattered from a zirconium oxide crystal.

The two patterns were remarkably similar, providing strong evidence for the wave nature of electrons.

Matter waves and atomic structure

de Broglie extended his matter wave concept to electrons within atoms, proposing that:

Electrons in atoms behave like standing waves, which encircle the nucleus in an integral number of whole wavelengths.

Understanding standing waves

Standing waves are waves confined between boundaries that vibrate in place rather than propagating. They possess:

• Nodes: Points that remain stationary
• Anti-nodes: Points that oscillate between maximum positive and negative displacements

The diagram shows standing waves with different numbers of wavelengths. When these linear standing waves are joined end-to-end to form circles, they resemble electron waves surrounding a nucleus.

Quantisation of electron waves

For electron waves to remain stable around a nucleus, they must satisfy a crucial condition: the wave must complete an integer number of wavelengths as it circles the nucleus. This ensures the wave's beginning and end are in phase. If the wavelength is not an integer multiple, the beginning and end will be out of phase, causing destructive interference that destroys the wave.

Therefore:

• Electrons in the first energy shell ($n = 1$) complete one full wavelength
• Electrons in the second energy shell ($n = 2$) complete two full wavelengths
• Electrons in the third energy shell ($n = 3$) complete three full wavelengths

And so forth for higher energy levels.

Explaining Bohr's model

de Broglie's electron wave model provides theoretical justification for Bohr's atomic model:

Explaining stability of electron orbits:

In de Broglie's model, electrons are standing waves rather than moving charges. Since standing waves don't propagate, they remain stable and don't emit electromagnetic radiation. This explains Bohr's first postulate about stable electron orbits.

Deriving quantisation of angular momentum:

de Broglie's model enables mathematical derivation of Bohr's third postulate. Starting with the requirement for an integer number of wavelengths:

$2\pi r_n = n\lambda$

Substituting de Broglie's equation $\lambda = \frac{h}{mv}$:

$2\pi r_n = \frac{nh}{mv}$

Rearranging:

$mv(2\pi r_n) = nh$

$mvr_n = \frac{nh}{2\pi}$

This is exactly Bohr's quantisation condition for angular momentum, now derived from fundamental wave principles rather than assumed empirically.

Modern applications of matter waves

de Broglie's matter wave theory forms the foundation of quantum mechanics and enables several important technologies:

Electron microscopy:

Electron microscopes exploit the wave nature of electrons to create images of extremely small objects. Since we cannot resolve objects smaller than the wavelength used to observe them, and electron wavelengths are much shorter than visible light wavelengths, electron microscopes can image structures far beyond the capability of optical microscopes. This technology has proven invaluable in medical and biological sciences, allowing detailed study of bacteria, viruses, and cellular structures.

Neutron diffraction:

At facilities like the OPAL nuclear reactor in Lucas Heights (near Sydney), scientists use the wave properties of neutrons to create diffraction patterns. These patterns help investigate material properties and detect stress damage in machine components.

Single particle interference experiments

Despite confirming wave properties of particles, a profound question remained: what exactly is "waving"? How do individual particles, detected at specific points, produce interference patterns?

Taylor's single photon experiment (1909)

GI Taylor performed a groundbreaking experiment using light so dim that only one photon existed in the apparatus at any moment. Remarkably, an interference pattern still developed on the photographic plate. This result made sense for a wave interpretation - a wave spreads through space and passes through both slits simultaneously, even at low intensity. But how could a single particle pass through both slits at once?

Tonomura's single electron experiment (1989)

Akira Tonomura and colleagues in Japan performed the definitive single-electron interference experiment.

Experimental setup:

• An electron gun fired individual electrons toward an electron biprism (which functions like a double slit)
• Electrons were sent one at a time - never more than one electron in the apparatus simultaneously
• A detector screen recorded where each electron arrived

Results:

Initially, with only a few electrons, arrival positions appeared random. However, as more electrons accumulated, a clear interference pattern emerged - alternating bands of high and low electron density. The pattern matched predictions from the double-slit formula $d\sin\theta = m\lambda$ (where $m = 0, 1, 2, ...$), using the electron's de Broglie wavelength.

Critical finding:

Since electrons arrived individually, they couldn't interfere with other electrons. Each single electron somehow interacted with both slits simultaneously, interfering with itself.

The probability wave interpretation

These experiments led to the modern quantum mechanical interpretation: particles are described by probability waves. The wave represents the probability of finding the particle at different locations. When these probability waves pass through both slits, they interfere, creating the observed pattern of high and low probability regions. Individual particles still arrive at discrete points (exhibiting particle behaviour), but their collective distribution follows the wave interference pattern.

Experiments testing this interpretation revealed a startling result: when detectors determined which slit an electron passed through, the interference pattern disappeared. The act of observation fundamentally changed the electron's behaviour - it could no longer simultaneously pass through both slits. This remains one of the most puzzling and profound aspects of quantum mechanics.