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# 8.3: Two-State System

Consider a system in which the time-independent Hamiltonian possesses two eigenstates, denoted

 (752) (753)

Suppose, for the sake of simplicity, that the diagonal matrix elements of the interaction Hamiltonian, , are zero:

 (754)

The off-diagonal matrix elements are assumed to oscillate sinusoidally at some frequency :

 (755)

where and are real. Note that it is only the off-diagonal matrix elements that give rise to the effect which we are interested in--namely, transitions between states 1 and 2.

For a two-state system, Equation (749) reduces to

 (756) (757)

where , and it is assumed that . Equations (756) and (757) can be combined to give a second-order differential equation for the time variation of the amplitude :

 (758)

Once we have solved for , we can use Equation (757) to obtain the amplitude . Let us look for a solution in which the system is certain to be in state 1 at time . Thus, our initial conditions are and . It is easily demonstrated that the appropriate solutions are

 (759) (760)

The probability of finding the system in state 1 at time is simply . Likewise, the probability of finding the system in state 2 at time is . It follows that

 (761) (762)

Equation (761) exhibits all the features of a classic resonance. At resonance, when the oscillation frequency of the perturbation, , matches the frequency , we find that

 (763) (764)

According to the above result, the system starts off at in state . After a time interval , it is certain to be in state 2. After a further time interval , it is certain to be in state 1, and so on. In other words, the system periodically flip-flops between states 1 and 2 under the influence of the time-dependent perturbation. This implies that the system alternatively absorbs and emits energy from the source of the perturbation.

The absorption-emission cycle also take place away from the resonance, when . However, the amplitude of oscillation of the coefficient is reduced. This means that the maximum value of is no longer unity, nor is the minimum value of zero. In fact, if we plot the maximum value of as a function of the applied frequency, , then we obtain a resonance curve whose maximum (unity) lies at the resonance, and whose full-width half-maximum (in frequency) is . Thus, if the applied frequency differs from the resonant frequency by substantially more than then the probability of the system jumping from state 1 to state 2 is 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 . Clearly, the weaker the perturbation (i.e., the smaller becomes), the narrower the resonance.