Understand matter waves the easy way, with plain English intuition, the de Broglie wavelength, electron diffraction as evidence, worked examples and an auto marked practice test. VCE Physics Units 3 and 4.
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Light can act like a stream of tiny particles, so here is the twist that finishes the story: matter can act like a wave. An electron, which you normally picture as a tiny ball, can spread out and ripple just like light does. Every moving particle carries its own wavelength, and there is one neat rule that tells you exactly how long it is.
Every moving particle has a wavelength
The rule is short. Take any moving particle, work out its momentum (mass times velocity), and divide Planck’s constant by it. That gives you the particle’s de Broglie wavelength.
A small momentum gives a long wavelength that is easy to notice.
A large momentum gives a tiny wavelength that is impossible to notice.
This is why an electron, which is feather light, has a wavelength big enough to measure, while a cricket ball does not. The cricket ball still has a wavelength, but its momentum is so huge that the wavelength is far smaller than anything we could ever detect.
The proof: firing electrons at a crystal
If matter really is wavy, it should diffract, which is the spreading and overlapping that only waves do. So physicists tested it. They fired a beam of electrons at a thin crystal and looked at the screen behind it.
What appeared was a pattern of rings, exactly the kind of pattern a wave makes when it diffracts. Particles fired through gaps would simply land in a clump. The rings could only happen if the electrons were behaving as waves. This is direct evidence that matter has wave properties.
The blue arrows are the electron beam heading in. The grey grid of dots is the crystal the electrons pass through. On the right, the red diffraction rings are the giveaway: a clump would mean particles, but rings mean waves.
See it for yourself
Switch to the de Broglie view and watch the electron settle into a standing wave wrapped around the nucleus. Only a whole number of wavelengths fits each ring, which is exactly why the electron can only sit on certain orbits.
Here is a clean idea that examiners love. The de Broglie rule connects wavelength and momentum the same way for both light and matter. Rearranged, it says momentum equals Planck’s constant divided by wavelength.
So if a photon of light and an electron of matter happen to have the same wavelength, they must also have the same momentum. It does not matter that one has mass and one does not. The wavelength alone fixes the momentum.
How to actually solve one
The method is a short, repeatable recipe.
Find the momentum of the particle. For matter that is p=mv. For a wavelength question it may be given directly.
To find the wavelength, divide Planck’s constant by the momentum: λ=ph.
To find the momentum from a wavelength, flip it around: p=λh.
Keep your powers of ten lined up, and remember the electron mass is 9.1×10−31 kg.
A quick sanity check: electrons should come out around 10−10 m, about the size of an atom. If your answer is wildly bigger or smaller, recheck the powers of ten.
Lock it in with active recall
Cover the answer and say each one out loud before you flip. Rate yourself honestly — the cards you find hard come back sooner, the ones you know are spaced further out.
Active recall
Answer from memory first, then flip. Rate yourself and each card returns on a spaced schedule (1 → 3 → 7 → 16 days).
Write the de Broglie wavelength of a moving particle.
λ=ph=mvh — Planck’s constant divided by momentum.
Rearrange to find momentum from wavelength.
p=λh. This relation holds for both light and matter.
What experiment proves matter has wave properties?
Firing a beam of electrons at a thin crystal produces diffraction rings — only waves can diffract, so matter must be wavy.
Why do we never notice the wave nature of a tennis ball?
Its momentum is enormous, and λ=h/p, so its wavelength is unimaginably small — far too tiny to detect.
A photon and an electron have the same wavelength. What else is equal?
Their momentum, since p=λh depends only on wavelength. It does not matter that one has mass and one does not.
Roughly how big is an electron’s de Broglie wavelength?
Around 10−10 m, about the size of an atom — which is why electrons diffract off crystal lattices.
Recall · Diffraction and Interference
When is diffraction most pronounced?
When the wavelength is comparable to or larger than the gap or spacing — a large λ/w ratio.
Recall · The Photoelectric Effect
What is the energy of a photon of frequency f?
E=hf, with h=6.63×10−34 J s — the same Planck’s constant that fixes the de Broglie wavelength.
See the recipe in action in the Worked Examples tab, then test yourself in Try It.
Worked examples
Worked Example 1The wavelength of a moving electron
An electron moves at 2.0×106 m/s. Find its de Broglie wavelength, using the electron mass m=9.1×10−31 kg and h=6.63×10−34 J s.
1
The de Broglie wavelength of any moving particle is Planck's constant divided by its momentum, and the momentum is just mass times velocity.
λ=ph=mvh
2
Work out the momentum on the bottom first.
mv=(9.1×10−31)(2.0×106)=1.82×10−24 kg m/s
3
Now divide Planck's constant by that momentum.
λ=1.82×10−246.63×10−34=3.6×10−10 m
Answer
λ=3.6×10−10 m
Worked Example 2Same wavelength, same momentum
A photon and an electron each have a wavelength of 5.0×10−10 m. Find the momentum of each, using h=6.63×10−34 J s.
1
Momentum and wavelength are tied together by the same rule for both light and matter. Rearrange the de Broglie relation to make momentum the subject.
p=λh
2
The wavelength is the same for both, so put the numbers in once.
p=5.0×10−106.63×10−34=1.3×10−24 kg m/s
3
Because momentum depends only on the wavelength here, the photon and the electron come out identical.
pphoton=pelectron=1.3×10−24 kg m/s
Answer
p=1.3×10−24 kg m/s for both
Practice questions
Practice test
Try it yourself
6 questions, 8 marks
Choose your answers, then submit to see your score and the full worked solutions.
Multiple choice is marked for you, just like Exam 2 Section A.
Q1.The de Broglie wavelength of a moving particle is given by:
1mark
Need a hint?
Wavelength is Planck's constant divided by momentum.
Show worked solution
The de Broglie wavelength is λ=ph, Planck's constant divided by the momentum. A larger momentum means a smaller wavelength, which is why fast, heavy objects have wavelengths far too tiny to notice.
Q2.Firing a beam of electrons at a thin crystal produces a pattern of rings on a screen. This is direct evidence that:
1mark
Need a hint?
Only waves produce diffraction patterns.
Show worked solution
Diffraction, the spreading and overlapping that makes a ring pattern, is something only waves do. Seeing it from a beam of electrons is direct evidence that matter has wave properties, exactly as de Broglie predicted.
Q3.Why do we never notice the wave nature of an everyday object like a tennis ball?
1mark
Need a hint?
Wavelength is h/p, and a large momentum makes it tiny.
Show worked solution
An everyday object has an enormous momentum compared with an electron, and wavelength is λ=h/p. A huge momentum on the bottom makes the wavelength unimaginably small, so the wave behaviour is there but is far too tiny to ever observe.
Q4.A particle has a de Broglie wavelength of 2.0×10−10 m. Using h=6.63×10−34 J s, its momentum is closest to:
1mark
Need a hint?
Use p=λh.
Show worked solution
p=λh=2.0×10−106.63×10−34=3.3×10−24 kg m/s. Multiplying instead of dividing, or mixing up the powers of ten, gives the other options.
Q5.An electron has a momentum of 3.0×10−24 kg m/s. Taking h=6.63×10−34 J s, find its de Broglie wavelength. Show your working.
3marks
Work this on paper. The worked solution appears once you submit.
Show worked solution
The de Broglie wavelength is
λ=ph=3.0×10−246.63×10−34.
Dividing gives
λ=2.2×10−10 m.
Q6.Electrons and X-rays are passed through the same crystal and produce diffraction patterns with the same spacing. This is because the electrons and X-rays used have the same:
1mark
Need a hint?
The amount of diffraction depends on the ratio of wavelength to gap size. Same crystal and same pattern spacing means what about the wavelengths?
Show worked solution
Diffraction depends on the ratio of wavelength to the slit (gap) spacing. With the same crystal and the same pattern spacing, the electrons and X-rays must have the same de Broglie wavelength (option A), even though their masses, speeds and energies all differ.
VCAA 2025 Physics Exam, Section A Q15
Frequently asked questions
What is a matter wave?
It is the wave behaviour that every moving particle has. Just as light can act like a stream of particles, a moving particle like an electron can spread out and overlap like a wave, with a wavelength set by its momentum.
Why don't we see people or cars behaving like waves?
Their momentum is enormous, and wavelength is Planck's constant divided by momentum. That makes the wavelength so unimaginably small that the wave behaviour can never be noticed in everyday life.
How do we know electrons really are waves?
When a beam of electrons is fired at a thin crystal it produces a diffraction pattern of rings on a screen, and only waves can diffract. That pattern is direct experimental proof that matter has wave properties.
