The Particle in a Box: Where Quantised Levels Come From · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Trapped waves can only sing certain notes

Science · Chemistry · ages 18-19
Name ______________________   Date ____________

  1. How many interior nodes does the n = 4 level have?

    Answer: ______________

  2. Why do only integer quantum numbers survive in the box?

    • Electrons simply prefer whole numbers
    • The box fills with charge in integer steps
    • The walls pin the wave to zero at each end

  3. How does the energy of level 3 compare with the ground level?

    • 3 times the ground energy
    • 9 times the ground energy
    • 27 times the ground energy

  4. Two similar dyes differ only in chain length. Which absorbs the longer wavelength?

    • The shorter chain
    • The longer chain
    • Both absorb the same colour

  5. The box is made twice as long. What happens to the level gaps?

    • They shrink
    • They grow wider
    • They stay exactly the same

  6. The ground energy of a box is 2 units. What is the energy of level 3 in the same units?

    Answer: ______________

  7. A box calculation matches a dye peak exactly. A student trusts it to four figures. What is the mistake?

    • Box results are always whole numbers of energy units
    • Dyes never absorb visible light at all
    • The box ignores uneven spacing and repulsion, so it gives trends only

  8. A student claims n = 0 is the ground state with zero energy. What is wrong?

    • The n = 0 wave is zero everywhere and cannot be normalised
    • Zero energy states are reserved for spring-like traps
    • Nothing, rest energy is always zero
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Answer key

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

Trapped waves can only sing certain notes W1-mt_rOCXlEi324-s1

  1. 3 · Level n always carries n minus 1 interior nodes.
  2. The walls pin the wave to zero at each end · Pinned ends admit only whole half waves, hence integer labels.
  3. 9 times the ground energy · Box energies grow with n squared, and 3 squared is 9.
  4. The longer chain · Longer chains have smaller jumps, and smaller jumps mean longer wavelengths.
  5. They shrink · A longer box squeezes every gap smaller.
  6. 18 · Level 3 carries 9 times the ground energy, and 9 times 2 is 18.
  7. The box ignores uneven spacing and repulsion, so it gives trends only · Real chains are messier than the box, so exact matches deserve suspicion.
  8. The n = 0 wave is zero everywhere and cannot be normalised · Throwing out n = 0 is what keeps every level a real wave.
Worksheet · LightMySky