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L1-09 · Unknown is not the same as quantum

In the two-path experiment with light, some spots stay dark because the amplitudes from the two paths cancel. Ordinary chances can only add up, but amplitudes can cancel. When a coin is hidden under a cup, you do not know how it landed. Not knowing which way a coin landed is different: the coin already landed one way.

Level 1 icon Level 1 · Grades 4–6 · about 40 minutes
Big idea. In the two-path experiment with light, some spots stay dark because the amplitudes from the two paths cancel. Ordinary chances can only add up, but amplitudes can cancel. When a coin is hidden under a cup, you do not know how it landed. Not knowing which way a coin landed is different: the coin already landed one way.
Before this lesson. L0-08, L1-08
Grown-ups. The guide for adults is at the end of this page.

What you will learn

Story: The coin under the cup

Maya flips a coin and slaps a cup over it before anyone can see. “Heads or tails?” she asks. Leo shrugs. “I don’t know. My chance of guessing right is one half.” Maya lifts the cup. Tails. “It was tails the whole time,” she says. “I did not know either, but the coin already landed one way.”

They play 20 rounds. Leo guesses right 9 times, which is 9 out of 20, close to one half. His chance was one half because he did not know. Every time, the coin under the cup was already heads or already tails.

The next day, their teacher shows a picture of the two-path experiment. Light from one lamp shines through two narrow slits onto a wall. The wall shows stripes of light and dark. “In the two-path experiment with light, some spots stay dark because the amplitudes from the two paths cancel,” she says.

She gives them simple numbers for one dark spot. With only path 1 open, the chance of light landing there is 1/4. With only path 2 open, it is also 1/4. If chances just added, we would get 1/4 + 1/4 = 1/2. Chances are never below zero, so adding two chances that are above zero can never make zero. But the spot is dark.

In quantum physics, chances come from amplitudes. Amplitudes add first, then you square the total to get a chance. The amplitude from path 1 is 1/2 up, and the one from path 2 is 1/2 down. Then 1/2 up and 1/2 down add to 0, and 0 × 0 = 0. That is a dark spot. “Ordinary chances can only add up, but amplitudes can cancel,” says Maya. “The coin under the cup was just unknown. Not knowing which way a coin landed is different: the coin already landed one way.”

Left: two chances stack and can never make zero. Right: two amplitudes, one up and one down, add to zero, and zero squared is zero.

What the animation shows. On the left, two chance bars of 1/4 stack on top of each other, and the screen shows 1/4 + 1/4 = 1/2, never 0. On the right, one amplitude arrow points 1/2 up and the next points 1/2 down, so 1/2 up and 1/2 down add to 0. Then the screen shows 0 × 0 = 0. At the end, the words “chances add; amplitudes can cancel” appear.

Hands-on: Hidden coins and paper arrows

About 20 minutes.

You need:

Steps:

  1. Play the hidden-coin game with a partner. One person flips the coin and covers it with the cup. The other guesses, then lifts the cup.

  2. Play 20 rounds and keep a tally of right and wrong guesses. Write your count as a fraction, like 9 out of 20.

  3. Cut two paper strips, each 4 squares long. These are chance bars. A chance bar always stands up from the line, never down.

  4. Draw a line across a sheet of paper. Stand both chance bars on it, one on top of the other. How tall is the stack? Can you make it reach zero?

  5. Cut two more strips, each 4 squares long, and draw an arrowhead at one end of each. These are amplitude arrows. An arrow can point up or down.

  6. Put the first arrow pointing up from the line. Put the tail of the second arrow at the tip of the first, pointing down. Where does the second arrow end?

  7. Write one sentence: which pair could add to zero, the chance bars or the amplitude arrows?

Practice together

At one spot, path 1 gives an amplitude of 1/2 up and path 2 gives 1/2 down. Add the amplitudes, then square the total. Is the spot bright or dark?

Check your answer

1/2 up and 1/2 down add to 0. Then 0 × 0 = 0. The spot is dark.

Try it yourself

1. At another spot, both paths give an amplitude of 1/2 up. Add the amplitudes, then square. Compare your answer with 1/4 + 1/4.

Check your answer

1/2 up and 1/2 up add to 1 up, and 1 × 1 = 1. That is twice as much as 1/4 + 1/4 = 1/2. Where amplitudes add in step, the spot is extra bright.

2. Sam says, “Until we lift the cup, the coin is half heads and half tails.” What would you tell Sam?

Check your answer

The coin already landed one way, heads or tails. The one half is our chance of guessing right, because we do not know which. Lifting the cup just shows us the side that was already up.

Talk about it

Check yourself

1. Maya hides a coin under a cup. Which sentence is true?

Show the answer

C. The coin already landed one way, and we just do not know which.

Not knowing which way a coin landed is different: the coin already landed one way. The one half is about what we know, not about the coin.

2. Which pair can add to zero?

Show the answer

C. Amplitudes of 1/2 up and 1/2 down

Chances are never below zero, so adding two chances that are above zero can never make zero. An amplitude up and an equal amplitude down cancel.

3. Remember Lesson 3. Leo guesses the hidden coin right 9 times in 20 rounds. What is 9 out of 20? (Review of L1-03)

Show the answer

B. A frequency: what really happened in the rounds

9 out of 20 is what Leo really counted, so it is a relative frequency. The model’s chance of guessing right is one half.

Words to know

amplitude
A number with a size and a direction, up or down, like the height of a wave. Two amplitudes can cancel.
cancel
To add up to zero, like 1/2 up and 1/2 down.
two-path experiment
Light from one lamp reaches a wall through two narrow slits, making stripes of light and dark.
For grown-ups: about this lesson

Why it matters. The lesson separates two kinds of uncertainty. A hidden coin is ordinary missing information: a probability describes what we know, and the coin already has a definite face. In the two-path (two-slit) experiment, the chance of arrival is found by adding the amplitudes for the two paths first and then taking the absolute square, so the pattern with both paths open is not the sum of the two one-path patterns Feynman, 1966. Quantum states are described by amplitudes, and measurement probabilities are the squared magnitudes of amplitudes, unlike a classical probabilistic state, whose entries are themselves probabilities Basics of quantum information, n.d.. Preskill’s lecture notes develop the same rules for quantum information at an advanced level Preskill, n.d.. The full contrast, too long for the children’s reading limit, is: “If each bit of light had simply taken one path, and we just did not know which, adding the two chances could never give zero there.” The children’s text makes the same point with numbers: chances of 1/4 and 1/4 add to 1/2 and can never give 0, while amplitudes of 1/2 up and 1/2 down add to 0. The numbers 1/2 and 1/4 are idealized, as in a balanced two-path setup. Real amplitudes are complex numbers with a size and a phase; the up and down arrows are the simplest case, two amplitudes with opposite signs. In this lesson, chance means the same as probability in Lessons 2 and 3.

Watch for these mix-ups.

  • A coin under a cup is half heads and half tails until someone lifts the cup. (It already landed one way; the one half describes our missing information.)

  • Amplitudes are just chances with a new name. (Chances are never below zero; amplitudes can point opposite ways and cancel, and you square the total to get a chance.)

  • Dark stripes mean light was destroyed. (The light that is missing from the dark stripes shows up in the bright stripes.)

  • Any uncertainty is quantum. (Most everyday uncertainty, like a hidden coin, is ordinary missing information.)

To make it easier. Do the hidden-coin game and the paper strips only. Use strips 2 squares long: two chance bars that stack to 4 squares, and two arrows, 2 squares up and 2 squares down, that end back at the line.

To make it harder. Try the in-step case: two amplitude arrows of 1/2 up add to 1, and 1 × 1 = 1, which is twice 1/4 + 1/4. Talk about how a bright stripe can get more light than the two chances would give, while a dark stripe gets less.

Limits. Bright light also makes stripes in ordinary wave optics, so a single picture of stripes is not by itself the quantum point. The quantum point is that the same striped pattern still builds up when the light is so dim that it arrives one small bit at a time. Feynman describes this build-up for electrons that arrive one at a time, each landing in one place, with the stripes appearing in the pattern of many arrivals, and he says that what we learn about electrons applies also to photons of light Feynman, 1966. That is why the guide’s sentence about each bit of light taking one path matters: if that were the whole story, adding the two chances could never give a dark spot. The lesson does not say what light does between the lamp and the wall; questions about interpretation are beyond this level.

On the computer. Grown-ups can see where this idea leads later in Lab: Qubits and the Bloch sphere, written for older learners.

Sources for this guide. Feynman, 1966Basics of quantum information, n.d.Preskill, n.d.

Next: L1-10 · Keep a model testable

References
  1. Feynman, R. P. (1966). The Feynman Lectures on Physics. Volume III. https://www.feynmanlectures.caltech.edu/III_toc.html
  2. Basics of quantum information. (n.d.). https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information
  3. Preskill, J. (n.d.). Course Information for Physics 219/Computer Science 219 Quantum Computation (Formerly Physics 229). California Institute of Technology course page with lecture notes. https://www.preskill.caltech.edu/ph219/