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Next: Spin Magnetic Resonance Up: Time-Dependent Perturbation Theory Previous: Preliminary Analysis

Two-State System

Consider a system in which the time-independent Hamiltonian possesses two eigenstates, denoted
$\displaystyle H_0 \psi_1$ $\textstyle =$ $\displaystyle E_1 \psi_1,$ (1031)
$\displaystyle H_0 \psi_2$ $\textstyle =$ $\displaystyle E_2 \psi_2.$ (1032)

Suppose, for the sake of simplicity, that the diagonal elements of the interaction Hamiltonian, $H_1$, are zero: i.e.,
\begin{displaymath}
\langle 1\vert H_1\vert 1\rangle = \langle 2\vert H_1\vert 2\rangle = 0.
\end{displaymath} (1033)

The off-diagonal elements are assumed to oscillate sinusoidally at some frequency $\omega$: i.e.,
\begin{displaymath}
\langle 1\vert H_1\vert 2\rangle = \langle 2\vert H_1\vert 1\rangle^\ast = \gamma \hbar \exp({\rm i} \omega t),
\end{displaymath} (1034)

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

For a two-state system, Eq. (1028) reduces to

$\displaystyle {\rm i} \frac{dc_1}{dt}$ $\textstyle =$ $\displaystyle \gamma \exp\left[+{\rm i} (\omega-\omega_{21}) t\right] c_2,$ (1035)
$\displaystyle {\rm i} \frac{dc_2}{dt}$ $\textstyle =$ $\displaystyle \gamma \exp\left[-{\rm i} (\omega-\omega_{21}) t\right] c_1,$ (1036)

where $\omega_{21}=(E_2-E_1)/\hbar$. The above two equations can be combined to give a second-order differential equation for the time-variation of the amplitude $c_2$: i.e.,
\begin{displaymath}
\frac{d^2 c_2}{dt^2} + {\rm i} (\omega-\omega_{21}) \frac{dc_2}{dt}+\gamma^2 c_2=0.
\end{displaymath} (1037)

Once we have solved for $c_2$, we can use Eq. (1036) to obtain the amplitude $c_1$. Let us search for a solution in which the system is certain to be in state 1 (and, thus, has no chance of being in state 2) 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 to (1037) and (1036) are
$\displaystyle c_2(t)$ $\textstyle =$ $\displaystyle \left(\frac{-{\rm i} \gamma}{\Omega}\right)\exp\!\left[\frac{-{\rm i} (\omega-\omega_{21}) t}{2}\right]\sin(\Omega t),$ (1038)
$\displaystyle c_1(t)$ $\textstyle =$ $\displaystyle \exp\!\left[\frac{{\rm i} (\omega-\omega_{21}) t}{2}\right]
\cos(\Omega t)$  
    $\displaystyle - \left[\frac{{\rm i} (\omega-\omega_{21})}{2 \Omega} \right]\exp\!\left[\frac{{\rm i} (\omega-\omega_{21}) t}{2}\right]\sin(\Omega t),$ (1039)

where
\begin{displaymath}
\Omega = \sqrt{\gamma^2 + (\omega-\omega_{21})^2/4}.
\end{displaymath} (1040)

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

$\displaystyle P_1(t)$ $\textstyle =$ $\displaystyle 1-P_2(t),$ (1041)
$\displaystyle P_2(t)$ $\textstyle =$ $\displaystyle \left[\frac{\gamma^2}{\gamma^2 + (\omega-\omega_{21})^2/4}\right]
\sin^2(\Omega t).$ (1042)

This result is known as Rabi's formula.

Equation (1042) 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

$\displaystyle P_1(t)$ $\textstyle =$ $\displaystyle \cos^2(\gamma t),$ (1043)
$\displaystyle P_2(t)$ $\textstyle =$ $\displaystyle \sin^2(\gamma t).$ (1044)

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


next up previous
Next: Spin Magnetic Resonance Up: Time-Dependent Perturbation Theory Previous: Preliminary Analysis
Richard Fitzpatrick 2010-07-20