Quantum Mechanics 101 · Unit 2: Wave–Particle Duality

Quantum Academy · Chapter 6: Matter Waves: de Broglie's Audacious Symmetry

If light can be a particle, can a particle be a wave?

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Instructions. Work through these on paper before checking the answers. Use λ = h/p, p = mv, p = √(2mK), p = E/c = h/λ and d·sin θ = nλ. Take h = 6.626×10⁻³⁴ J·s, c = 2.998×10⁸ m/s, 1 eV = 1.602×10⁻¹⁹ joules, electron mass 9.109×10⁻³¹ kg, neutron mass 1.675×10⁻²⁷ kg, and 1 atomic mass unit = 1.661×10⁻²⁷ kg. Show units at every step. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Matter wave
2.De Broglie wavelength
3.Momentum
4.Radiation pressure
5.Compton scattering
6.Diffraction
7.Diffraction grating
8.Standing wave
9.Resolution
10.Thermal neutron
11.Fullerene
12.Probability amplitude

A.The quantity a matter wave describes; its squared size, |ψ|², gives the probability of finding the particle at each place.
B.A closed cage molecule of carbon; C₆₀, buckminsterfullerene, is a football-shaped cage of 60 carbon atoms about a nanometre across.
C.The spreading and patterning of a wave as it passes edges, openings or regularly spaced obstacles comparable in size to its wavelength.
D.The lengthening of an X-ray's wavelength when it scatters from an electron, explained in 1923 as a particle collision in which the photon carries momentum h/λ.
E.The push exerted by light on a surface that absorbs or reflects it, a consequence of photon momentum.
F.A regular array of lines, slits or rows of atoms that sends a wave into bright beams at angles where d·sin θ = nλ.
G.The wavelength of a particle's matter wave, λ = h/p, equal to Planck's constant divided by the particle's momentum.
H.The wave associated with any moving particle of matter, also called a de Broglie wave; its squared amplitude gives the probability of detecting the particle.
I.A neutron slowed to the thermal energy of its surroundings, about 0.025 eV and 2,200 m/s, with a wavelength near 0.18 nm.
J.The smallest separation at which an instrument can show two details as distinct; for a microscope, roughly half the wavelength used at best.
K.A wave pattern that oscillates in place without travelling, possible only when a whole number of half- or full wavelengths fits the confining space.
L.The quantity of motion an object carries and conserves in collisions: p = mv for slow bodies, p = E/c = h/λ for photons.

Part B Questions

1.
According to de Broglie, which property of a moving particle determines its wavelength?Circle one
1. AIts size
2. BIts electric charge
3. CIts momentum
4. DIts temperature alone
2.
Why did electron diffraction need a crystal rather than the ruled gratings used for light?Circle one
1. AElectrons can only be reflected by metals
2. BElectron wavelengths of about 0.1 to 0.2 nm match the spacing of atoms in a crystal, and are far smaller than the line spacing of any ruled grating
3. CHeating a crystal turns electrons into waves
4. DA crystal slows electrons until their wavelength becomes visible light
3.
What mainly limits the resolution of a modern transmission electron microscope?Circle one
1. AThe electron wavelength, which is about the size of an atom
2. BThe size of the electrons themselves, which blurs every image
3. CThe speed of the electrons, which can never exceed the speed of light
4. DImperfections, called aberrations, of the magnetic lenses that focus the beam
4.
Only light shows wave properties; particles of matter such as electrons and atoms never diffract.Circle one
TrueFalse
5.
In the 1999 C₆₀ experiment, each molecule was detected whole at a single place, and the diffraction pattern appeared only after many molecules had arrived.Circle one
TrueFalse
6.
If a slow electron's speed is doubled, its de Broglie wavelength is halved.Circle one
TrueFalse
7.
In de Broglie's explanation of Bohr's atom, the circumference of an allowed orbit holds a whole number of electron .Fill in the blank
8.
A photon's momentum equals its energy divided by the .Fill in the blank
9.
Explain why a thrown baseball shows no detectable wave behaviour even though de Broglie's relation applies to it.Short answer
10.
A friend says an electron's matter wave is the electron's material spread out like a ripple on a pond. Using evidence from diffraction experiments, explain what is wrong.Short answer
11.
An electron is accelerated from rest through 150 volts. Find its kinetic energy in joules, its momentum and its de Broglie wavelength. Check with the electron shortcut.Show your working
Working
Answer:
12.
A thermal neutron moves at 2,200 m/s. Find its de Broglie wavelength and its kinetic energy in electron-volts. Is it suitable for studying crystals whose atoms are about 0.2 nm apart?Show your working
Working
Answer:
13.
A C₆₀ molecule (mass 720 atomic mass units) travels at 220 m/s toward a grating whose slits are 100 nm apart. Find its de Broglie wavelength, the angle of the first-order diffraction peak, and how far that peak lies from the central peak on a detector 1.25 metres beyond the grating.Show your working
Working
Answer:
14.
A 5.0 milliwatt green laser (532 nm) shines on a perfectly absorbing target. Find the momentum of one photon, the number of photons arriving per second, and the force on the target.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

Braking an atom with light. A sodium atom (mass 3.82×10⁻²⁶ kg) leaves an oven at 600 m/s and flies head-on into a laser beam of wavelength 589 nm, tuned so that the atom absorbs its photons. (a) Find the atom's de Broglie wavelength. (b) Find the momentum of one photon and the change in the atom's speed each time it absorbs one. (c) Each absorbed photon is soon re-emitted in a random direction. Explain why absorptions slow the atom while emissions, on average, do not speed it up again, and estimate how many absorptions are needed to stop it. (d) Show that an atom moving at exactly the speed change found in (b) has a de Broglie wavelength equal to the laser's wavelength, and explain why this had to be so.

Working

Memory aid

Mnemonic Two hooks, one for the formula and one for the evidence. The formula: 'Push hard, wave small.' Read λ = h/p literally, with Planck's tiny h on top and the push, the momentum p, underneath: the harder an object pushes, the smaller its wave, which is why fast, heavy things look classical and slow, light things look quantum. The evidence, in the order nature confessed it: Every Atom Now Feels Giddy. Electrons (1927, Davisson and Germer, and Thomson), Atoms (1930, Estermann and Stern's helium), Neutrons (1936, crystal diffraction), Fullerenes (C₆₀, 1999), Giants (2,000-atom molecules in 2019 and 7,000-atom sodium clusters in 2026). Whenever a question asks whether something has a wave, walk the ladder: the answer is always yes, and the only real question is whether h/p is big enough to see.

In one line Everything with momentum has a wavelength, λ = h/p: atom-sized for electrons, absurdly tiny for baseballs, and confirmed from electrons to giant molecules.