3. Wave Nature of Matter

Wave Nature of Matter

For centuries, scientists often treated matter and waves as fundamentally different things.

Matter, such as an electron, was pictured as a particle with:

• mass
• momentum
• position
• velocity

Waves were described using properties such as:

• wavelength
• frequency
• diffraction
• interference

The development of quantum physics revealed that this distinction is not always valid.

Particles such as electrons can produce distinctly wave-like behaviour.

This idea is called the:

wave nature of matter

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From Light to Matter

Scientists first discovered that light could behave in apparently contradictory ways.

Light produces wave phenomena such as:

• interference
• diffraction
• polarization

But experiments such as the photoelectric effect showed that light can also behave as discrete packets called:

photons

A photon has energy:

E = hf

and momentum:

p = h/λ

This led to an important question:

If waves such as light can behave like particles, could particles such as electrons also behave like waves?

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de Broglie's Hypothesis

In 1924, French physicist Louis de Broglie proposed that particles have an associated wavelength.

He suggested:

λ = h/p

where:

• λ = de Broglie wavelength
• h = Planck's constant
• p = momentum

For a non-relativistic particle:

p = mv

Therefore:

λ = h/mv

This is the:

de Broglie equation

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Planck's Constant

Planck's constant is:

h = 6.63 × 10⁻³⁴ J·s

It is an extremely small number.

This is one reason matter-wave effects are usually noticeable only for very small particles such as:

• electrons
• neutrons
• atoms
• molecules

For ordinary macroscopic objects, the de Broglie wavelength is usually far too small to observe directly.

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Momentum and Wavelength

The de Broglie equation is:

λ = h/p

Therefore:

wavelength is inversely proportional to momentum

If momentum increases:

p ↑ → λ ↓

If momentum decreases:

p ↓ → λ ↑

For a particle of constant mass:

higher speed → greater momentum → shorter wavelength

and:

lower speed → smaller momentum → longer wavelength

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Mass and Wavelength

Using:

λ = h/mv

for particles moving at the same speed:

greater mass → shorter wavelength

Therefore, very massive objects have extraordinarily small matter wavelengths.

This helps explain why quantum wave behaviour is much easier to observe for electrons than for everyday objects.

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Worked Example: Electron Wavelength

An electron travels at:

2.0 × 10⁶ m/s

Electron mass:

m = 9.11 × 10⁻³¹ kg

Use:

λ = h/mv

First calculate momentum:

p = mv

p = (9.11 × 10⁻³¹)(2.0 × 10⁶)

p = 1.82 × 10⁻²⁴ kg·m/s

Now calculate wavelength:

λ = (6.63 × 10⁻³⁴)/(1.82 × 10⁻²⁴)

λ ≈ 3.64 × 10⁻¹⁰ m

Therefore:

λ ≈ 0.364 nm

This wavelength is comparable with distances between atoms in crystals.

That fact is extremely important for electron diffraction.

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Worked Example: Increasing Speed

Suppose the same electron doubles its speed.

Originally:

λ = h/mv

If:

v → 2v

then:

λ → λ/2

Therefore, doubling the speed approximately halves the de Broglie wavelength, provided the speed remains low enough that relativistic corrections are unnecessary.

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Worked Example: Finding Momentum

A particle has a de Broglie wavelength of:

5.0 × 10⁻¹⁰ m

Find its momentum.

Use:

λ = h/p

Rearrange:

p = h/λ

p = (6.63 × 10⁻³⁴)/(5.0 × 10⁻¹⁰)

p ≈ 1.33 × 10⁻²⁴ kg·m/s

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Matter Waves

The wave associated with a particle is often called a:

matter wave

or:

de Broglie wave

This does not mean an electron is simply a tiny vibrating water wave.

Matter waves are quantum-mechanical.

Their wave behaviour is connected to quantities such as:

• probability
• momentum
• interference
• diffraction

This eventually led to the development of the quantum mechanical:

wavefunction

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What Would Demonstrate Wave Behaviour?

If electrons behave like waves, they should be capable of producing phenomena normally associated with waves.

Two especially important phenomena are:

diffraction

and:

interference

Therefore, scientists searched for evidence that beams of electrons could produce diffraction patterns.

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Diffraction

Diffraction is the spreading or bending of waves when they encounter:

• openings
• edges
• regularly spaced structures

Strong diffraction effects occur when the dimensions of the structure are comparable to the wavelength.

For electrons, atomic crystals provide extremely small regularly spaced structures.

Their atomic spacing is often around:

10⁻¹⁰ m

This is comparable to electron de Broglie wavelengths.

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Crystals as Diffraction Structures

A crystal contains atoms arranged in a regular repeating pattern called a:

crystal lattice

The spacing between atomic planes can be comparable with the wavelength of moving electrons.

Therefore, a crystal can act somewhat like a three-dimensional diffraction structure for electron matter waves.

If electrons were purely classical particles, this wave-like diffraction behaviour would not be expected.

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Davisson-Germer Experiment

In 1927, Clinton Davisson and Lester Germer performed one of the most famous experiments confirming de Broglie's hypothesis.

They directed a beam of electrons toward a:

nickel crystal

The electrons scattered from the crystal surface.

Instead of producing only the distribution expected for classical particles, the electrons produced strong intensity maxima at particular angles.

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What Davisson and Germer Observed

The scattered electrons produced a pattern consistent with:

diffraction

Certain angles produced strong electron signals.

Other angles produced weaker signals.

This behaviour was similar to interference and diffraction produced by waves.

The measured wavelength agreed with the wavelength predicted using:

λ = h/p

This provided powerful experimental evidence for:

de Broglie's matter waves

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G. P. Thomson's Experiment

At around the same time, George Paget Thomson performed another important experiment.

He directed electrons through thin metal foils.

The electrons produced:

diffraction rings

on a detector.

These rings closely resemble diffraction patterns produced by other types of waves.

The results provided further evidence that:

electrons possess wave properties

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An Interesting Historical Connection

J. J. Thomson had earlier helped establish the electron as a:

particle

His son, G. P. Thomson, later demonstrated the electron's:

wave behaviour

Both descriptions turned out to be important.

This became part of the broader concept of:

wave-particle duality

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Bragg's Law and Electron Diffraction

Diffraction from crystal planes can be described using:

Bragg's law

nλ = 2d sinθ

where:

• n = diffraction order
• λ = wavelength
• d = spacing between crystal planes
• θ = Bragg angle

Electron diffraction therefore allows the wavelength of electrons to be measured experimentally.

The result can then be compared with:

λ = h/p

Agreement between these two approaches supports de Broglie's hypothesis.

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Worked Example: Electron Diffraction

Suppose first-order diffraction occurs from crystal planes separated by:

d = 0.20 nm

at an angle:

θ = 30°

For first order:

n = 1

Use:

nλ = 2d sinθ

λ = 2(0.20 nm)sin30°

Since:

sin30° = 0.5

λ = 2(0.20)(0.5)

λ = 0.20 nm

The measured electron wavelength is therefore:

2.0 × 10⁻¹⁰ m

This can be compared with the wavelength predicted from the electron's momentum.

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Interference

Matter waves can also produce:

interference

When wave amplitudes combine, they can produce:

• constructive interference
• destructive interference

Constructive interference increases detection probability in some locations.

Destructive interference decreases it in others.

This produces characteristic patterns.

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Electron Double-Slit Experiment

One of the most striking demonstrations of matter waves involves electrons passing through two narrow slits.

If electrons behaved only like classical particles, we might expect two main regions behind the two slits.

Instead, after many electrons are detected, an:

interference pattern

can appear.

This is characteristic of wave behaviour.

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Individual Electron Detections

An especially important feature of the double-slit experiment is that electrons can be sent through the apparatus:

one at a time

Each electron is detected at a particular location.

This appears particle-like.

However, after many detections, the accumulated pattern reveals:

wave-like interference

This demonstrates that quantum objects do not fit neatly into classical categories.

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Particle Behaviour

Electrons show particle-like properties because they can:

• carry electric charge
• possess momentum
• transfer energy in localized interactions
• produce individual detector events
• collide with other particles

For example, an electron detector records an electron at a particular location rather than detecting a tiny continuous classical wave spread across the entire screen.

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Wave Behaviour

Electrons show wave-like properties because they can produce:

• diffraction
• interference
• wavelength-dependent patterns

Their wavelength is related to momentum through:

λ = h/p

Therefore, neither the simple classical particle picture nor the simple classical wave picture is sufficient on its own.

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Wave-Particle Duality

The idea that quantum objects can display both particle-like and wave-like behaviour is called:

wave-particle duality

Light can display:

wave behaviour + particle behaviour

Matter can also display:

particle behaviour + wave behaviour

This was one of the major conceptual changes introduced by quantum physics.

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Classical Particle vs Quantum Particle

A classical particle is usually imagined as:

• occupying a definite position
• following a definite trajectory
• possessing definite momentum
• moving along a predictable path

Quantum particles cannot always be described this way.

Instead, quantum mechanics uses a:

wavefunction

to describe the quantum state.

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The Wavefunction

The wavefunction is usually represented by:

ψ

The wavefunction itself is not directly interpreted as an ordinary physical wave like a water wave.

Instead:

|ψ|²

is related to the probability density for detecting a particle.

This connects wave behaviour with the probability of particle detection.

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Probability Waves

In quantum mechanics, an electron's wave behaviour helps determine:

where the electron is likely to be detected

For example, an interference experiment produces regions where:

• detection is more likely
• detection is less likely

The individual detection remains localized.

But the distribution of many detections follows a wave-like pattern.

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Why We Do Not See Matter Waves for Everyday Objects

Every object has a de Broglie wavelength.

This includes:

• electrons
• atoms
• people
• cars
• footballs
• planets

But for macroscopic objects, the wavelength is extraordinarily small.

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Worked Example: Football

Suppose a football has:

m = 0.50 kg

v = 20 m/s

Momentum:

p = mv

p = (0.50)(20)

p = 10 kg·m/s

Now:

λ = h/p

λ = (6.63 × 10⁻³⁴)/10

λ = 6.63 × 10⁻³⁵ m

This wavelength is unimaginably small.

It is far too small to produce observable diffraction with ordinary structures.

Therefore, a football behaves extremely well as a:

classical particle

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Electron vs Football

Compare:

Electron

Possible wavelength:

~10⁻¹⁰ m

This is comparable with atomic spacing.

Diffraction can therefore be observed.

Football

Possible wavelength:

~10⁻³⁵ m

No practical slit or crystal structure is remotely small enough to reveal such diffraction.

Therefore:

quantum behaviour is not absent from large objects—it is simply normally unobservable at everyday scales.

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Accelerating Electrons Through a Voltage

Electrons are often accelerated using an electric potential difference.

If an electron begins approximately from rest and is accelerated through voltage V:

electrical energy gained = kinetic energy

Therefore:

eV = ½mv²

where e is the elementary charge.

Since:

p = mv

we can write:

KE = p²/(2m)

Therefore:

p = √(2meV)

Substitute this into de Broglie's equation:

λ = h/√(2meV)

This is useful for electron diffraction experiments.

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Worked Example: Electron Accelerated Through a Voltage

Suppose an electron is accelerated through:

150 V

Use:

λ = h/√(2meV)

where:

h = 6.63 × 10⁻³⁴ J·s

m = 9.11 × 10⁻³¹ kg

e = 1.60 × 10⁻¹⁹ C

V = 150 V

This gives approximately:

λ ≈ 1.00 × 10⁻¹⁰ m

or:

λ ≈ 0.100 nm

Again, this is comparable with atomic spacings in crystals.

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Voltage and Electron Wavelength

From:

λ = h/√(2meV)

we can see:

λ ∝ 1/√V

Therefore:

higher accelerating voltage → greater momentum → shorter wavelength

For example, if the voltage is multiplied by 4:

λ becomes half as large

provided relativistic effects remain negligible.

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Electron Microscopes

The small de Broglie wavelength of fast electrons has an important practical application:

electron microscopy

Optical microscopes use visible light.

Visible wavelengths are approximately:

400–700 nm

Electrons can have much shorter de Broglie wavelengths.

Shorter wavelengths can potentially reveal much smaller structures.

This allows electron microscopes to achieve extremely high resolution.

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Why Wavelength Affects Resolution

Diffraction limits the ability of imaging systems to distinguish nearby objects.

In general:

shorter wavelength → potentially better resolution

Electron microscopes therefore use the wave nature of electrons to investigate structures much smaller than those easily resolved using visible-light microscopes.

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Electron Diffraction in Materials Science

Electron diffraction is also used to investigate:

• crystal structures
• atomic spacing
• material phases
• crystal orientation
• defects

The diffraction pattern contains information about the arrangement of atoms inside the material.

Thus de Broglie's hypothesis became an important practical tool as well as a fundamental quantum idea.

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Neutron Diffraction

Electrons are not the only particles that show wave behaviour.

Neutrons also have de Broglie wavelengths.

Neutron diffraction can be used to study:

• crystal structures
• magnetic structures
• atomic positions

This confirms that matter waves are not unique to electrons.

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Atoms and Molecules

Experiments have demonstrated interference and diffraction using increasingly large particles, including:

• atoms
• molecules

This reinforces the idea that the de Broglie relationship applies generally to matter.

The transition between obvious quantum behaviour and everyday classical behaviour is therefore not simply:

small things quantum, large things classical

The situation also depends on interactions with the environment and the ability to maintain quantum coherence.

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de Broglie and the Bohr Model

de Broglie's idea also helped explain why Bohr's allowed electron states might occur.

Imagine an electron's matter wave fitting around an orbit.

For a stable standing wave:

circumference = whole number of wavelengths

Therefore:

2πr = nλ

where:

n = 1, 2, 3, ...

Only certain wavelengths fit exactly.

This provides a wave-based way of understanding why only certain states are allowed.

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Standing Waves and Quantization

Consider a wave around a circular path.

If the circumference contains:

1 wavelength

or:

2 wavelengths

or:

3 wavelengths

the wave can join smoothly with itself.

But a non-integer number of wavelengths would not form the same stable standing-wave pattern.

This connects:

wave behaviour → allowed states → quantization

The modern quantum model is more sophisticated than this simple picture, but the standing-wave analogy is useful.

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From de Broglie to Quantum Mechanics

de Broglie's hypothesis had a major influence on the development of quantum theory.

If particles have wave properties, scientists needed an equation describing those waves.

This helped inspire Erwin Schrödinger to develop:

wave mechanics

The resulting Schrödinger equation became one of the foundations of quantum mechanics.

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Particle and Wave Descriptions

Quantum mechanics incorporates aspects of both descriptions.

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Neither Classical Picture Is Complete

It is tempting to ask:

"Is an electron really a particle or really a wave?"

Quantum physics shows that neither classical picture completely describes an electron.

An electron is a:

quantum object

Depending on the experiment, its behaviour may reveal characteristics that resemble classical:

• particles
• waves

The experimental arrangement determines which properties can be observed.

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Matter Waves Are Not Ordinary Waves

Matter waves should not be confused with:

• sound waves
• water waves
• waves on strings

These classical waves involve physical disturbances in a medium or field.

Quantum matter waves describe the state and probability behaviour of particles.

This is why the modern interpretation is more subtle than saying:

"an electron is literally a tiny wave."

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Evidence for de Broglie's Hypothesis

Strong evidence includes:

Electron diffraction

Electrons produce diffraction patterns when scattered from crystals.

Electron interference

Electrons can form interference patterns.

Agreement with λ = h/p

Experimentally measured electron wavelengths agree with de Broglie's prediction.

Neutron and atom diffraction

Other particles also show wave-like behaviour.

Electron microscopy

The wave properties of electrons are routinely exploited in technology.

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A Powerful Scientific Prediction

de Broglie's hypothesis is an important example of scientific reasoning.

The logic was approximately:

light behaves as waves

but:

light also behaves as particles

therefore perhaps:

matter behaves as particles

but could also:

behave as waves

This was a bold theoretical prediction.

Experiments later confirmed it.

That sequence:

hypothesis → prediction → experiment → evidence

is a powerful example of how physics develops.

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Worked Example: Comparing Two Electrons

Electron A has momentum:

p

Electron B has momentum:

2p

For electron A:

λ_A = h/p

For electron B:

λ_B = h/(2p)

Therefore:

λ_B = λ_A/2

Electron B has twice the momentum and half the wavelength.

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Worked Example: Comparing Two Particles

Particle A:

m = m

v = v

Particle B:

m = 2m

v = v

Since:

λ = h/mv

Particle B has twice the momentum.

Therefore:

λ_B = λ_A/2

The more massive particle has the shorter de Broglie wavelength when both move at the same speed.

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Worked Example: Finding Speed

An electron has wavelength:

4.0 × 10⁻¹⁰ m

Find its speed.

Use:

λ = h/mv

Rearrange:

v = h/(mλ)

Substitute:

v = (6.63 × 10⁻³⁴) / [(9.11 × 10⁻³¹)(4.0 × 10⁻¹⁰)]

v ≈ 1.82 × 10⁶ m/s

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Problem-Solving Strategy

For de Broglie calculations:

Identify what is known

Look for:

• mass
• velocity
• momentum
• wavelength
• accelerating voltage

Calculate momentum if necessary

For non-relativistic motion:

p = mv

Use de Broglie's equation

λ = h/p

or:

λ = h/mv

Check units

Momentum:

kg·m/s

Wavelength:

m

Check reasonableness

Small particles often have measurable wavelengths.

Large everyday objects should produce extremely tiny wavelengths.

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Common Mistakes

Mistake: Saying only light has wave-particle duality.
Matter particles such as electrons also exhibit wave behaviour.

Mistake: Saying de Broglie's equation is λ = hp.
The correct relationship is λ = h/p.

Mistake: Saying greater momentum gives a greater wavelength.
Wavelength is inversely proportional to momentum.

Mistake: Forgetting to calculate momentum first.
For non-relativistic motion, use p = mv.

Mistake: Thinking electron diffraction proves electrons are ordinary classical waves.
Electrons are quantum objects exhibiting wave-like diffraction while still producing localized detections.

Mistake: Saying electrons disappear into waves.
Individual electrons are still detected as localized events.

Mistake: Saying matter waves are only theoretical.
Electron, neutron, atom, and molecular diffraction provide experimental evidence.

Mistake: Saying everyday objects have no de Broglie wavelength.
They do, but the wavelength is extraordinarily small.

Mistake: Assuming λ = h/mv is valid at all speeds.
At speeds approaching the speed of light, relativistic momentum must be used.

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Key Terms

de Broglie hypothesis — proposal that matter has an associated wavelength determined by its momentum.

de Broglie wavelength — wavelength associated with a particle, given by λ = h/p.

Matter wave — quantum wave associated with a material particle.

Momentum — quantity of motion; for non-relativistic particles, p = mv.

Planck's constant — fundamental constant h = 6.63 × 10⁻³⁴ J·s.

Wave-particle duality — quantum principle that matter and radiation can display both particle-like and wave-like behaviour.

Diffraction — spreading or interference of waves caused by interaction with structures comparable in size to their wavelength.

Interference — combination of waves producing regions of reinforcement and cancellation.

Crystal lattice — regular repeating arrangement of atoms in a crystal.

Davisson-Germer experiment — experiment demonstrating electron diffraction from a nickel crystal.

Bragg's law — relationship nλ = 2d sinθ used to describe diffraction from crystal planes.

Wavefunction — mathematical description of a quantum state.

Probability density — quantity related to the probability of detecting a quantum particle in a particular region.

Electron diffraction — wave-like scattering of electrons that produces diffraction patterns.

Standing wave — wave pattern produced when waves combine so that stable nodes and antinodes form.

Quantum object — object described by quantum mechanics rather than completely by classical particle or wave models.

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Key Takeaways

• de Broglie proposed that matter has wave properties.
• The de Broglie wavelength is λ = h/p.
• For non-relativistic particles, λ = h/mv.
• Greater momentum produces a shorter de Broglie wavelength.
• Greater speed produces a shorter wavelength for a particle of fixed mass.
• Greater mass produces a shorter wavelength for particles travelling at the same speed.
• Electrons can have wavelengths comparable with atomic spacings in crystals.
• Crystal lattices can therefore produce electron diffraction.
• The Davisson-Germer experiment provided strong evidence for electron matter waves.
• Electron diffraction measurements agree with de Broglie's predicted wavelength.
• Electrons can also produce interference patterns.
• Individual electrons are detected at localized positions even though many detections form wave-like patterns.
• This behaviour is part of wave-particle duality.
• Matter waves are not ordinary mechanical waves.
• Modern quantum mechanics connects the wavefunction with probabilities of particle detection.
• All matter has a de Broglie wavelength.
• Macroscopic objects have wavelengths so tiny that their wave behaviour is normally impossible to observe.
• Electron wavelengths can be much shorter than visible-light wavelengths.
• Electron microscopy uses this property to achieve very high resolution.
• de Broglie's hypothesis helped lead to the development of modern quantum mechanics.

Check Your Understanding

1. State de Broglie's hypothesis.
2. Write the equation for de Broglie wavelength.
3. What does p represent in λ = h/p?
4. How does wavelength change when particle momentum increases?
5. Calculate the momentum of an electron moving at 3.0 × 10⁶ m/s.
6. Use your answer to Question 5 to calculate its de Broglie wavelength.
7. What happens to an electron's wavelength if its momentum doubles?
8. What happens to its wavelength if its momentum is halved?
9. Why are matter-wave effects easier to observe for electrons than for footballs?
10. What is diffraction?
11. Why are crystals useful for demonstrating electron diffraction?
12. Describe the Davisson-Germer experiment.
13. What observation from the experiment provided evidence for electron wave behaviour?
14. How did the measured electron wavelength support de Broglie's hypothesis?
15. Explain how the electron double-slit experiment demonstrates both particle-like and wave-like behaviour.
16. What is meant by wave-particle duality?
17. Calculate the de Broglie wavelength of a 0.20 kg ball travelling at 5.0 m/s.
18. Explain why the wave nature of the ball in Question 17 would not normally be observable.
19. Why can electron microscopes achieve much higher resolution than ordinary optical microscopes?
20. Compare the classical particle description and wave description of an electron, and explain why modern quantum mechanics requires a description that goes beyond either classical picture alone.