Quantum Mechanics 101 · Unit 2: Wave–Particle Duality

Quantum Academy · Chapter 7: The Double-Slit Experiment

One particle at a time, two paths, and the only mystery

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Instructions. Work these on paper before checking the answers. Use Δy ≈ λL/d, d sin θ = mλ, λ = h/p, P = |ψ₁ + ψ₂|² and D² + V² ≤ 1. Take h = 6.626×10⁻³⁴ J·s, electron mass 9.109×10⁻³¹ kg and 1 eV = 1.602×10⁻¹⁹ joules. Use the small-angle approximation and 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.Fringe
2.Fringe spacing
3.Path difference
4.Probability amplitude
5.Interference term
6.Indistinguishable alternatives
7.Which-path information
8.Complementarity
9.Fringe visibility
10.Entanglement
11.Decoherence
12.Quantum eraser

A.The difference in distance from the two slits to a point on the screen, d sin θ; whole numbers of wavelengths give bright fringes.
B.An experiment that recovers interference in sorted subsets of the data by measuring partner systems in a way that cannot reveal the path.
C.Any physical record, read or unread, that could reveal which slit a particle went through.
D.Ways an event can happen that leave no record anywhere of which one occurred; their amplitudes add before squaring.
E.A joint state of two systems, such as a particle and a which-path marker, that cannot be described as separate states of each.
F.Bohr's principle of 1927 that wave-like interference and particle-like path knowledge cannot both be fully displayed in one experimental arrangement.
G.One of the bright or dark bands of an interference pattern.
H.The loss of interference that occurs when a system becomes entangled with its environment, which carries away which-path information.
I.The distance between neighbouring bright fringes on the screen, Δy ≈ λL/d for small angles.
J.The contrast of an interference pattern, (highest − lowest) ÷ (highest + lowest) probability; 1 for perfect fringes, 0 for none.
K.A complex number assigned to each way an event can happen; the probability is its squared magnitude.
L.The part of |ψ₁ + ψ₂|² beyond |ψ₁|² + |ψ₂|²: twice the product of the amplitudes' sizes times the cosine of their phase difference.

Part B Questions

1.
In Feynman's comparison, what do electrons have in common with bullets?Circle one
1. ABoth produce interference fringes when both slits are open
2. BBoth arrive at the detector in whole, identical lumps
3. CBoth have probabilities that simply add when both slits are open
4. DBoth deliver energy continuously, like a water wave
2.
A double-slit setup produces fringes 6.0 mm apart. The slit separation is doubled and the screen is moved to three times its original distance. What is the new fringe spacing?Circle one
1. A36 mm
2. B12 mm
3. C4.0 mm
4. D9.0 mm
3.
A which-path detector at the slits records each electron's slit on a hard drive, but the file is deleted unread before anyone looks. What does the screen show?Circle one
1. ANo fringes, because a physical record distinguishing the paths existed, read or not
2. BFringes, because the record was deleted before anyone could read it
3. CFringes for exactly half of the electrons, since no record was ever seen
4. DFringes that reappear the moment the file is deleted, as in a quantum eraser
4.
In Tonomura's 1989 experiment, the fringes formed because electrons in the beam interfered with one another.Circle one
TrueFalse
5.
Opening a second slit can reduce the number of particles arriving at certain points on the screen.Circle one
TrueFalse
6.
In the 2007 realization of Wheeler's delayed-choice experiment, the choice whether to recombine the two paths was made so that no light-speed signal could have told the photon about it before it entered the interferometer, and the results still agreed with ordinary quantum mechanics.Circle one
TrueFalse
7.
For indistinguishable alternatives, quantum mechanics says to add the probability first and square afterwards.Fill in the blank
8.
Bohr's name for the principle that interference and which-path knowledge cannot both be fully displayed in one experimental arrangement is .Fill in the blank
9.
Using evidence from real experiments, explain why it is wrong to say that each electron splits in half at the slits.Short answer
10.
Hot C₇₀ molecules lose their interference fringes even though no detector watches the slits. Explain what removes the interference and why no observer is required.Short answer
11.
Green laser light of wavelength 532 nm passes through two slits 0.10 mm apart. Find the fringe spacing on a screen 1.5 metres away, and the distance from the central bright fringe to the third bright fringe.Show your working
Working
Answer:
12.
A student shines a 633 nm helium-neon laser through an unknown double slit onto a wall 2.5 metres away and measures 40 mm from the central bright fringe to the tenth bright fringe on one side. What is the slit separation?Show your working
Working
Answer:
13.
In Tonomura's experiment, 50,000-volt electrons travel at about 1.24×10⁸ m/s, roughly 41 percent of the speed of light, and arrive at about 1,000 per second. Find the average distance between successive electrons, the time each spends crossing the 1.5-metre instrument, and the fraction of the time that any electron is inside it.Show your working
Working
Answer:
14.
At one point on a screen, slit 1 alone gives a probability density of 4 units and slit 2 alone gives 1 unit. With both slits open and nothing recording the path, find the largest and smallest possible values, the bullet-rule value, and the fringe visibility. What is the largest distinguishability a which-path detector could then have without lowering the visibility below this value?Show your working
Working
Answer:

Part C Mad scientist challenge bonus

Feynman's lamp. Electrons of 600 eV (wavelength 50.1 pm) pass two slits 272 nm apart and land on a screen 1.0 metre away. (a) Find the fringe spacing and the angle between neighbouring fringes. (b) To see which slit each electron uses, you scatter light off it; roughly, light can tell the slits apart only if its wavelength is shorter than the slit separation. Using 200 nm ultraviolet light, estimate the sideways kick a scattered photon can give an electron, as an angle, and the resulting smear on the screen. Do the fringes survive? (c) Repeat for gentle 1,000 nm infrared light. Can that light tell the slits apart? (d) Show that the ratio of kick angle to fringe angle is simply d divided by the light's wavelength, and explain why this makes Feynman's lamp a no-win device. (e) Why do physicists now say the kick is not the deepest reason the fringes vanish?

Working

Memory aid

Mnemonic Think DOTS, because dots are all a double-slit screen ever shows you. D is for Dots: every particle lands whole, as one dot, never half. O is for Order of operations: add the amplitudes first, square second, and the cross term paints the stripes. T is for Tape measure: Δy ≈ λL/d, so a longer wavelength or a longer throw spreads the stripes and a wider slit separation squeezes them. S is for Snitch: if anything anywhere, a photon, a gas molecule, an atom's internal state, keeps a record of the path, the stripes are gone, whether or not anyone reads it. Then seal it with a rhyme: if the world can know, the stripes won't show. Face any double-slit question by walking the letters in order: where do the dots land, what adds, how far apart, and who snitched?

In one line Each particle lands as one dot, but indistinguishable paths add amplitudes into stripes; leave any record of the path anywhere, and the stripes vanish.