The Wavefunction and the Born Probability Rule · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Where the particle might be

Science · Quantum & Modern Physics · ages 18-20
Name ______________________   Date ____________

  1. The squared magnitude of the position wave function gives the probability density.

    Circle one:   True   False

  2. What is the wave function psi?

    • Something you can see and touch directly
    • An encoding of where the particle might be found
    • The particle's exact position

  3. Probability density is constant across an interval. How do you find the probability of detecting the particle there?

    • Multiply the density by the interval width
    • Divide the interval width by the density
    • Integrate psi itself across the interval

  4. Identically prepared electrons pass through two open slits without which-slit information. Only one electron is in the apparatus at a time, yet many detections build stripes. What does this show?

    • Interference does not require several electrons together
    • Every electron must land at a position predicted exactly in advance
    • The interference pattern requires several electrons in flight together

  5. For a normalized wave function, what must the total probability over all space equal?

    • Zero
    • The particle mass
    • One

  6. The numerical value of psi at a point is i in the chosen units. Which calculation gives the probability density there?

    • i² = -1, so the density is -1
    • |i|² = 1, so the density is 1
    • psi = i, so the density is imaginary

  7. Two equal-width intervals have constant probability densities. Density in A is much higher than in B. Over many identically prepared particles, which interval is expected to collect more detections?

    • They must collect equal totals
    • A is expected to collect more
    • B is expected to collect more

  8. A student adds the two single-slit probability patterns to predict the pattern with both slits open and no which-slit information. Which correction is needed?

    • Add the probabilities, then square that probability sum
    • Choose one single-slit pattern and discard the other
    • Add the amplitudes, then take the squared magnitude
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Where the particle might be W1-mt_elzmbBzmKE-s1

  1. True · The Born rule uses |psi|², the squared magnitude. The ordinary square psi² can be complex or negative.
  2. An encoding of where the particle might be found · Psi is a mathematical description used to predict measurement outcomes, including where the particle may be detected.
  3. Multiply the density by the interval width · For constant density, the area under the density curve is a rectangle: density times interval width.
  4. Interference does not require several electrons together · Since only one electron is present at a time, the pattern does not depend on interference between different electrons. Repeated trials still build the distribution predicted from the combined two-slit amplitude.
  5. One · Normalization makes the integral of position probability density over all space equal one.
  6. |i|² = 1, so the density is 1 · The magnitude of i is one, so its squared magnitude is one. Squaring i itself gives minus one, which cannot be probability density.
  7. A is expected to collect more · Each interval probability is its constant density times its width. Equal widths make the higher-density interval A more probable.
  8. Add the amplitudes, then take the squared magnitude · Interference comes from combining amplitudes before applying the Born rule. Adding the separate probabilities omits interference.
Worksheet · LightMySky