The Finite Well and Wavefunction Penetration · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Leaky walls: the finite well and its fading tails

Science · Quantum & Modern Physics · ages 20-21
Name ______________________   Date ____________

  1. What sets how fast the wavefunction fades outside a finite well?

    • The gap between the particle energy and the wall height
    • The width of the well
    • The material of the walls

  2. A bound particle sits in a finite well. What shape does its wavefunction take just outside the wall?

    • A continued sine wave
    • Instant zero at the edge
    • An exponential fade with no oscillation

  3. A finite well holds infinitely many bound states.

    Circle one:   True   False

  4. How does a finite-well wave at the edge differ from an infinite-well wave?

    • It stops dead like the infinite well
    • It reflects with double height
    • It leaks past the edge instead of stopping dead

  5. At the boundary of a finite well, what must match between inside and outside?

    • Only the energy
    • The wavefunction and its slope
    • Nothing: the wave restarts freely

  6. Why can a finite well hold only a limited number of bound states?

    • Higher states spill over the rim, so only a few fit below it
    • The well shrinks as energy grows
    • Nodes are forbidden in finite wells

  7. A particle sits far below the rim, with a large shortfall. What does its tail look like?

    • Longer tail reaching further
    • No change in the tail
    • Faster fade, shorter tail

  8. You sketch the second bound state of a finite well. What must your drawing show?

    • No nodes and hard stops at the walls
    • One node inside plus fading tails outside
    • Two nodes and zero tails
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Answer key

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

Leaky walls: the finite well and its fading tails W1-mt_Vs4pWnXW7l-s1

  1. The gap between the particle energy and the wall height · A taller shortfall means a faster fade and a shorter tail.
  2. An exponential fade with no oscillation · In the forbidden zone the wave fades exponentially and never oscillates.
  3. False · States past the rim are unbound, so the ladder ends.
  4. It leaks past the edge instead of stopping dead · Finite walls cannot stop a quantum wave cold, so it leaks through.
  5. The wavefunction and its slope · Both the wave and its slope join smoothly, and that picks the energies.
  6. Higher states spill over the rim, so only a few fit below it · Each rung climbs higher, and rungs past the rim are not bound.
  7. Faster fade, shorter tail · A taller shortfall speeds the fade, so the tail shortens.
  8. One node inside plus fading tails outside · Level two carries one node, with oscillation inside and fading tails outside.
Worksheet · LightMySky