A quantum system starts in energy eigenstate |m⟩. A time-varying perturbation H'(t) = V₀ cos(ωt) is applied. The probability of transitioning to state |n⟩ is largest when which condition holds?
AThe perturbation amplitude V₀ is very large, regardless of frequency
BThe matrix element ⟨n|H'|m⟩ is non-zero AND the perturbation frequency ω is close to the Bohr frequency ω_{nm} = (E_n − E_m)/ℏ
CThe perturbation frequency ω is much larger than the Bohr frequency ω_{nm}
DThe system has remained in state |m⟩ for a long time before the perturbation is applied
Two conditions are needed for large transition probability: (1) a non-zero matrix element ⟨n|H'|m⟩ coupling the initial and final states, and (2) resonance — ω ≈ ω_{nm}. At resonance, the oscillating phase factor e^{i(ω_{nm} − ω)t'} in the transition amplitude integral becomes slowly varying and the integral accumulates coherently over time, giving transition probability growing as t². Off-resonance, the integrand oscillates rapidly and averages nearly to zero. Option A misses the resonance condition; large amplitude alone is insufficient.
Question 2 Multiple Choice
In first-order time-dependent perturbation theory, what physically happens when the perturbation frequency is far from resonance (ω ≪ ω_{nm})?
AThe transition probability grows linearly with time
BThe system undergoes an instantaneous transition to state |n⟩
CThe oscillating phase factor in the transition amplitude causes the integrand to average nearly to zero, producing negligible transition probability
DThe perturbation shifts the energy levels rather than inducing transitions
The transition amplitude involves the integral ∫₀ᵗ ⟨n|H'(t')|m⟩ e^{i(ω_{nm} − ω)t'} dt'. When ω is far from ω_{nm}, the phase factor e^{i(ω_{nm} − ω)t'} oscillates rapidly and the positive and negative contributions nearly cancel — the integral stays small regardless of how long the perturbation acts. This is why radio waves don't drive optical transitions and vice versa: frequency matching (energy conservation) is required for coherent accumulation of transition amplitude.
Question 3 True / False
Time-dependent perturbation theory addresses the same fundamental question as time-independent perturbation theory — finding corrected energy levels — but for Hamiltonians that vary with time.
TTrue
FFalse
Answer: False
These two theories address fundamentally different questions. Time-independent perturbation theory finds corrected energy eigenvalues and eigenstates when a static perturbation modifies a known Hamiltonian. Time-dependent perturbation theory asks: given a system initially in one eigenstate of H₀, what is the probability of finding it in a *different* eigenstate after a time-varying perturbation acts? The second is a question about transitions between states, not corrections to energy levels.
Question 4 True / False
At exact resonance (ω = ω_{nm}), the first-order transition probability grows with time because the phase factor in the transition amplitude integral becomes slowly varying, allowing coherent accumulation.
TTrue
FFalse
Answer: True
At resonance, ω_{nm} − ω = 0, so e^{i(ω_{nm} − ω)t'} = 1 — the integrand is constant rather than oscillating. The integral grows linearly with t, making the transition probability grow as t² (|amplitude|²). This coherent accumulation is the mathematical signature of resonance, and it is the mechanism behind stimulated absorption and emission, NMR spin flipping, and all coherent quantum drives. The t² growth is only an approximation valid at short times; at longer times the first-order approximation breaks down.
Question 5 Short Answer
Explain why resonance — the match between perturbation frequency and Bohr frequency — is essential for large transition probabilities in time-dependent perturbation theory.
Think about your answer, then reveal below.
Model answer: The transition amplitude involves integrating the matrix element times an oscillating phase factor e^{i(ω_{nm} − ω)t'}. When ω ≠ ω_{nm}, this phase oscillates rapidly and the positive and negative contributions cancel, keeping the amplitude near zero no matter how long the perturbation acts. When ω ≈ ω_{nm}, the phase factor is nearly constant and the integral grows linearly with time — coherent accumulation. The transition probability (amplitude squared) grows as t². Physically, resonance corresponds to the perturbation delivering energy quanta that match the level spacing, satisfying energy conservation for the transition.
This is the quantum mechanical foundation of spectroscopy and coherent control: only perturbations tuned to the right frequency drive transitions. The resonance condition is essentially energy conservation expressed in the time-domain through phase coherence.