Time-Dependent Perturbation Theory and Fermi's Golden Rule · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

How fast one state becomes another

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

  1. Which expression gives the golden-rule rate in its weak-coupling continuum regime?

    • Gamma=(2pi/hbar)|M| rho(E_res)
    • Gamma=(2pi/hbar)|M|² rho(E_res)
    • Gamma=(2pi/hbar)|M|²/rho(E_res)

  2. What does first-order time-dependent perturbation theory integrate to obtain a transition amplitude?

    • The coupling matrix element with its relative-energy phase
    • The final energy alone, without a coupling or relative-energy phase
    • The initial probability alone, without information about the perturbation

  3. A constant golden-rule decay rate is an exact description of any isolated pair of quantum levels.

    Circle one:   True   False

  4. For a constant coupling g=0.02 acting from 0 to T=3 at resonance, take hbar=1 in consistent units. What is the first-order amplitude?

    • -0.06i
    • 0.0036
    • -0.02i

  5. Two cases satisfy the golden-rule assumptions and have the same nonzero M. With rho_A(E_res)>0 and rho_B(E_res)=3rho_A(E_res), what is Gamma_B/Gamma_A?

    • 1/3
    • 1
    • 3

  6. A narrow interval of width 0.30 eV around E_res contains 72 final states. Assuming nearly constant density there, estimate rho(E_res).

    • 21.6 states/eV
    • 240 states/eV
    • 72.0 states/eV

  7. Within the golden-rule approximation, rho(E_res)=0 but M is nonzero. Ravi obtains a nonzero rate by counting final states at other energies. How should Ravi correct the calculation?

    • Use rho at E_res; this golden-rule contribution is zero.
    • Use the count at every energy; this golden-rule contribution stays nonzero.
    • Replace rho by |M|; the missing final states no longer affect the rate.

  8. An atom emits a photon while dropping from E_i=9 eV to E_f=4 eV. Neglect recoil. Which photon energy belongs to the energy-conserving final states?

    • 4 eV
    • 13 eV
    • 5 eV
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Answer key

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

How fast one state becomes another W1-mt_zDSASGF7SC-s1

  1. Gamma=(2pi/hbar)|M|² rho(E_res) · The rate contains the squared magnitude of M multiplied by the final-state density per unit energy.
  2. The coupling matrix element with its relative-energy phase · Integrate V_fi(t) exp[i(E_f-E_i)t/hbar] over the time window, then multiply by -i/hbar.
  3. False · The steady-rate approximation needs a suitable continuum and time regime. An isolated pair does not supply that continuum.
  4. -0.06i · At resonance a=-igT/hbar=-i×0.02×3=-0.06i. The number 0.0036 is |a|², not the amplitude.
  5. 3 · Nonzero M and positive rho_A make Gamma_A positive. Cancel the common prefactor and |M|²; the rate ratio is rho_B/rho_A=3.
  6. 240 states/eV · Divide the state count by the energy width: 72/0.30=240 states/eV. Multiplying instead gives the numerical slip 21.6.
  7. Use rho at E_res; this golden-rule contribution is zero. · The rule uses the density at the energy-conserving value. Its zero value makes this contribution zero, regardless of a nonzero M.
  8. 5 eV · Conserve the combined atom and photon energy: 4 eV+E_ph=9 eV, so E_ph=5 eV.
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