Quantum tunnelling and the transmission probability · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Through the wall: tunnelling and its exponential rule

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

  1. In an isolated nucleus undergoing alpha decay, the alpha particle's energy is below the top of the Coulomb barrier. What explains its escape?

    • A classical path over the Coulomb barrier
    • Tunnelling through the Coulomb barrier
    • A disappearance of the alpha particle's electric charge

  2. A stationary bound-state tail has nonzero exterior density. Which evidence would instead demonstrate transmission through a finite barrier?

    • Outgoing transmitted flux in an allowed region beyond the barrier
    • Nonzero density just inside the forbidden region
    • Nonzero exterior density in a stationary bound-state tail without escape

  3. At fixed bias and local electronic states, increasing an STM's vacuum gap makes its tunnelling current increase.

    Circle one:   True   False

  4. An STM with a sharp tip can resolve atomic-scale features. Why should its image not be treated as geometric height alone?

    • Current depends on gap width and local electronic states
    • Current depends only on geometric height
    • Current is measured by touching each surface atom with the tip

  5. Use the illustrative model T(L)=exp(-aL), with a fixed and positive. If T(L)=0.0001, what does the model predict when the width is halved?

    • 0.0002
    • 0.0001
    • 0.0100

  6. An electron and a much heavier proton have the same energy E below U0. They meet rectangular barriers with the same height U0 and width. Which has greater transmission probability?

    • Both particles, because equal energies give equal probabilities
    • The proton, because its greater mass reduces barrier attenuation
    • The electron, because its smaller mass reduces attenuation

  7. A student says elastic tunnelling through a static barrier breaks energy conservation because E is below the barrier top. What corrects this claim?

    • Total energy stays fixed during elastic tunnelling through a static barrier
    • The particle borrows extra energy and keeps it
    • The static barrier supplies the missing energy whenever the particle reaches its edge

  8. Use the small-transmission alpha-escape model: decay rate ≈ attempt rate × transmission per attempt. Attempt rate stays fixed while transmission becomes one quarter as large. What happens to half-life?

    • One quarter of the initial half-life
    • Four times the initial half-life
    • Equal to the initial half-life
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Answer key

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Through the wall: tunnelling and its exponential rule W1-mt_tLdPJKx8Id-s1

  1. Tunnelling through the Coulomb barrier · The alpha particle can be transmitted through the Coulomb barrier while its energy remains below the barrier top.
  2. Outgoing transmitted flux in an allowed region beyond the barrier · Nonzero bound-state density outside a well is penetration without escape. Transmission involves outgoing particle flux in a classically allowed region beyond a finite barrier.
  3. False · Under these controls, a wider vacuum gap reduces electron transmission and therefore reduces tunnelling current.
  4. Current depends on gap width and local electronic states · STM measures tunnelling current. Gap width affects that current, but local electronic states also affect the signal. Image contrast is therefore not geometric height alone.
  5. 0.0100 · Halving L takes the square root of T in this model. sqrt(0.0001)=0.01, which is 100 times the original probability.
  6. The electron, because its smaller mass reduces attenuation · With E, U0 and width fixed, increasing mass increases κ=√[2m(U0-E)]/ℏ and reduces transmission. The electron has the greater transmission probability.
  7. Total energy stays fixed during elastic tunnelling through a static barrier · Elastic scattering from a time-independent potential conserves total E. Transmission does not require a temporary energy loan or a classical path over the barrier.
  8. Four times the initial half-life · At fixed attempt rate, quartering transmission quarters decay rate in this model. Half-life is inversely proportional to decay rate, so it becomes about four times as long.
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