The Path Integral and the Sum Over Histories · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Path amplitudes and the classical limit

Science · Quantum & Modern Physics · ages 23-24
Name ______________________   Date ____________

  1. What histories enter the position-space path integral with fixed endpoint positions and times?

    • All admissible histories joining the fixed positions and times
    • Only stationary histories joining the fixed positions and times
    • Only paths of minimum action joining those fixed endpoints

  2. For real action S, which phase factor appears in the real-time path integral?

    • exp(-S/ℏ)
    • exp(iS/h)
    • exp(iS/ℏ)

  3. At one screen point, coherent slit amplitudes 5b and -5b give an ideal dark fringe because their sum is zero.

    Circle one:   True   False

  4. At a screen point, two coherent slit amplitudes are 3a and -a, with a nonzero. What is their sum?

    • 4a; add magnitudes despite the opposite phases
    • 2a; add amplitudes with their relative minus sign
    • 0; opposite phases cancel unequal magnitudes

  5. Under suitable semiclassical conditions, why can a neighborhood of a stationary-action path give a leading contribution?

    • Nearby phases agree to first order around stationary action
    • The action is zero along every stationary classical path
    • The spatial route is shortest for every stationary path

  6. For E<V in part of a finite static barrier, how does tunnelling fit the path-integral account?

    • Transmission requires each history to borrow energy
    • Transmission excludes histories crossing the barrier
    • Transmission can be nonzero in the full propagator

  7. In the model λ=fT, keep f fixed and increase transmission T from 0.01 to 0.02. What happens to decay rate λ and half-life ln(2)/λ?

    • The rate doubles; the half-life halves
    • The rate halves; the half-life doubles
    • The rate doubles; the half-life doubles

  8. Which situation can require more care than an approximation based on one isolated classical path?

    • A large action makes one path exact
    • Several stationary paths contribute
    • Two stationary paths make K undefined
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Answer key

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Path amplitudes and the classical limit W1-mt_QRLheflB1O-s1

  1. All admissible histories joining the fixed positions and times · The integral includes admissible histories joining the fixed endpoints. Stationary paths enter later when constructing an approximation.
  2. exp(iS/ℏ) · The phase angle is S/ℏ, not S/h. For real S, exp(iS/ℏ) has modulus one. exp(-S/ℏ) is not the real-time phase factor.
  3. True · The total amplitude is 5b+(-5b)=0, so its squared modulus is zero.
  4. 2a; add amplitudes with their relative minus sign · 3a+(-a)=2a. Opposite phases do not completely cancel unequal magnitudes.
  5. Nearby phases agree to first order around stationary action · At δS=0, the first-order phase change vanishes. Nearby contributions can reinforce while rapidly varying contributions largely cancel.
  6. Transmission can be nonzero in the full propagator · The evaluated quantum propagator can transmit through a classically forbidden region. Energy is conserved, and real-action phase factors are not individually damped.
  7. The rate doubles; the half-life halves · T doubles, so λ=fT doubles. Half-life is inversely proportional to λ and therefore halves.
  8. Several stationary paths contribute · Several stationary contributions may need combining. Turning points can need a different local approximation. The full path integral does not thereby fail.
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