12.2: Two-State System
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Consider a system in which the time-independent Hamiltonian possesses two eigenstates, denoted H0ψ1=E1ψ1,H0ψ2=E2ψ2.
For a two-state system, Equation ([e13.12]) reduces to
idc1dt=γexp[+i(ω−ω21)t]c2,idc2dt=γexp[−i(ω−ω21)t]c1,
c2(t)=(−iγΩ)exp[−i(ω−ω21)t2]sin(Ωt)
c1(t)=exp[i(ω−ω21)t2]cos(Ωt)−[i(ω−ω21)2Ω]exp[i(ω−ω21)t2]sin(Ωt)
where Ω=√γ2+(ω−ω21)2/4
Now, the probability of finding the system in state 1 at time t is simply P1(t)=|c1(t)|2. Likewise, the probability of finding the system in state 2 at time t is P2(t)=|c2(t)|2. It follows that P1(t)=1−P2(t),P2(t)=[γ2γ2+(ω−ω21)2/4]sin2(Ωt).
Equation ([e13.25]) exhibits all the features of a classic resonance . At resonance, when the oscillation frequency of the perturbation, ω, matches the frequency ω21, we find that
P1(t)=cos2(γt),P2(t)=sin2(γt).
The absorption-emission cycle also takes place away from the resonance, when ω≠ω21. However, the amplitude of the oscillation in the coefficient c2 is reduced. This means that the maximum value of P2(t) is no longer unity, nor is the minimum of P1(t) zero. In fact, if we plot the maximum value of P2(t) as a function of the applied frequency, ω, then we obtain a resonance curve whose maximum (unity) lies at the resonance, and whose full-width half-maximum (in frequency) is 4γ. Thus, if the applied frequency differs from the resonant frequency by substantially more than 2γ then the probability of the system jumping from state 1 to state 2 is always very small. In other words, the time-dependent perturbation is only effective at causing transitions between states 1 and 2 if its frequency of oscillation lies in the approximate range ω21±2γ. Clearly, the weaker the perturbation (i.e., the smaller γ becomes), the narrower the resonance.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)