Two-State System

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

$$H_0 \,\vert 1\rangle = E_1\, \vert 1\rangle,$$ (752)

$$H_0 \,\vert 2\rangle = E_2 \,\vert 2\rangle.$$ (753)

Suppose, for the sake of simplicity, that the diagonal matrix elements of the interaction Hamiltonian, $H_1$ , are zero:

$$\langle 1\vert\,H_1\,\vert 1\rangle = \langle 2\vert\,H_1\,\vert 2\rangle = 0.$$ (754)

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

$$\langle 1\vert\,H_1\,\vert 2\rangle = \langle 2\vert\,H_1\,\vert 1\rangle^\ast = \gamma \exp(\,{\rm i}\, \omega\,t),$$ (755)

where $\gamma$ and $\omega$ 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

$${\rm i} \,\hbar\, \frac{d c_1}{dt} = \gamma \exp[+{\rm i}\, (\omega-\omega_{21})\,t\,]\,c_2,$$ (756)

$${\rm i}\,\hbar\, \frac{d c_2}{dt} = \gamma \exp[-{\rm i}\, (\omega-\omega_{21})\,t\,]\,c_1,$$ (757)

where $\omega_{21} = (E_2 - E_1)/\hbar$ , and it is assumed that $t_0=0$ . Equations (756) and (757) can be combined to give a second-order differential equation for the time variation of the amplitude $c_2$ :

$$\frac{d^2 c_2}{dt^2} + {\rm i}\,(\omega-\omega_{21})\,\frac{d c_2}{dt} + \frac{\gamma^2}{\hbar^2} \,c_2 = 0.$$ (758)

Once we have solved for $c_2$ , we can use Equation (757) to obtain the amplitude $c_1$ . Let us look for a solution in which the system is certain to be in state 1 at time $t=0$ . Thus, our initial conditions are $c_1(0) = 1$ and $c_2(0) = 0$ . It is easily demonstrated that the appropriate solutions are

$$c_2(t) = \frac{-{\rm i}\, \gamma/\hbar} {[\gamma^2/\hbar^2 + (\omega-\omeg... .../2]\,\sin\left([\gamma^2/\hbar^2+(\omega-\omega_{21})^{\,2}/4]^{1/2}\,t\right),$$ (759)

$$c_1(t)= \exp[\,{\rm i}\,(\omega-\omega_{21})\,t/2]\,\cos\left( [\gamma^2/\hbar^2+(\omega-\omega_{21})^{\,2}/4]^{1/2}\,t\right)$$

$$- \frac{{\rm i}\,(\omega-\omega_{21})/2 }{[\gamma^2/\hbar^2 + (\o... ...2]\,\sin\left( [\gamma^2/\hbar^2+(\omega-\omega_{21})^{\,2}/4]^{1/2}\,t\right).$$ (760)

The probability of finding the system in state 1 at time $t$ is simply $P_1(t) = \vert c_1\vert^{\,2}$ . Likewise, the probability of finding the system in state 2 at time $t$ is $P_2(t) = \vert c_2\vert^{\,2}$ . It follows that

$$P_2(t) = \frac{\gamma^2/\hbar^2}{ \gamma^2/\hbar^2 + (\omega-\omega_{21}... ..., \sin^2\left([\gamma^2/\hbar^2+ (\omega-\omega_{21})^{\,2}/4]^{1/2}\,t\right),$$ (761)

$$P_1(t) = 1 - P_2(t).$$ (762)

Equation (761) exhibits all the features of a classic resonance. At resonance, when the oscillation frequency of the perturbation, $\omega$ , matches the frequency $\omega_{21}$ , we find that

$$P_1(t) =\cos^2 (\gamma \,t / \hbar),$$ (763)

$$P_2(t) = \sin^2 (\gamma \,t/\hbar ).$$ (764)

According to the above result, the system starts off at $t=0$ in state $1$ . After a time interval $\pi \,\hbar/2\,\gamma$ , it is certain to be in state 2. After a further time interval $\pi \,\hbar/2\,\gamma$ , 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 $\omega\neq \omega_{21}$ . However, the amplitude of oscillation of the coefficient $c_2$ is reduced. This means that the maximum value of $P_2(t)$ is no longer unity, nor is the minimum value of $P_1(t)$ zero. In fact, if we plot the maximum value of $P_2(t)$ as a function of the applied frequency, $\omega$ , then we obtain a resonance curve whose maximum (unity) lies at the resonance, and whose full-width half-maximum (in frequency) is $4\,\gamma/\hbar$ . Thus, if the applied frequency differs from the resonant frequency by substantially more than $2\,\gamma/\hbar$ 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 $\omega_{21} \pm 2\,\gamma/\hbar$ . Clearly, the weaker the perturbation (i.e., the smaller $\gamma$ becomes), the narrower the resonance.