The Finite Well and Wavefunction Penetration · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Bound states in a finite square well

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

  1. Which quantities set the amplitude decay length outside a finite square well?

    • The mass and the gap U0 minus E
    • The well width and the slope
    • The well width and exterior potential U0

  2. A finite-square-well bound state has E below the constant exterior potential U0. What is its exterior wavefunction?

    • A constant-amplitude sine wave
    • A sharp cutoff to zero at the edge
    • An outward-decaying exponential

  3. A square well of fixed finite width and depth can have infinitely many bound states for a particle of fixed finite mass.

    Circle one:   True   False

  4. How does a finite-square-well bound-state wavefunction differ at an edge from an infinite-well wavefunction?

    • Zero at the edge and zero outside
    • Nonzero at the edge and constant outside
    • Nonzero at the edge and decaying outside

  5. At a finite-square-well edge, particle mass is the same on both sides and there is no singular potential spike. What must match?

    • Only the wavefunction value
    • The wavefunction and its slope
    • Only the wavefunction derivative

  6. Compare finite and infinite square wells with the same width, mass and interior potential. The finite well has a second bound state. How does its energy compare with the infinite well's second level?

    • Below
    • Equal
    • Above

  7. At one amplitude decay length ℓ outside a finite-square-well edge, what is the probability density divided by its value at the edge?

    • It is 1/e of the edge density
    • It is unchanged from the edge density
    • It is 1/e² of the edge density

  8. A finite square well has at least two bound states. Which features belong in a sketch of its second bound-state wavefunction?

    • No interior nodes and abrupt stops at both walls
    • One interior node and two decaying exterior tails
    • Two interior nodes and constant exterior amplitudes
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Answer key

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Bound states in a finite square well W1-mt_Vs4pWnXW7l-s1

  1. The mass and the gap U0 minus E · The amplitude decay length is ℏ/√[2m(U0-E)], so it depends on particle mass and the exterior energy gap.
  2. An outward-decaying exponential · For E<U0 the exterior solutions are exponential. Normalisation excludes growth toward infinity, leaving decay away from each edge.
  3. False · Bound states require E<U0. This limits the interior wave number, so only finitely many states fit and match both exterior tails.
  4. Nonzero at the edge and decaying outside · An infinite wall requires ψ=0 at the edge. A finite-square-well bound state joins to a nonzero exterior tail that decays outward.
  5. The wavefunction and its slope · At an ordinary finite potential step with constant mass, both ψ and its first spatial derivative are continuous.
  6. Below · With width, mass and interior potential held fixed, each finite-well bound level lies below the corresponding infinite-well level.
  7. It is 1/e² of the edge density · At d=ℓ, the wavefunction magnitude is 1/e of its edge magnitude. Squaring gives the density ratio 1/e², about 0.135.
  8. One interior node and two decaying exterior tails · The ground state has no interior nodes, so the second bound state has one. Its exterior tails decay on both sides.
Worksheet · LightMySky