8.3: Two-State System

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

\( H_0 \,\vert 1\rangle\) \( = E_1\, \vert 1\rangle,\) \ref{752} \( H_0 \,\vert 2\rangle\) \( = E_2 \,\vert 2\rangle.\) \ref{753}

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

\( \langle 1\vert\,H_1\,\vert 1\rangle = \langle 2\vert\,H_1\,\vert 2\rangle = 0.\) \ref{754}

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

\( \langle 1\vert\,H_1\,\vert 2\rangle = \langle 2\vert\,H_1\,\vert 1\rangle^\ast = \gamma \exp(\,{\rm i}\, \omega\,t),\) \ref{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 \ref{749} reduces to

\( {\rm i} \,\hbar\, \frac{d c_1}{dt}\) \( = \gamma \exp[+{\rm i}\, (\omega-\omega_{21})\,t\,]\,c_2,\) \ref{756} \( {\rm i}\,\hbar\, \frac{d c_2}{dt}\) \( = \gamma \exp[-{\rm i}\, (\omega-\omega_{21})\,t\,]\,c_1,\) \ref{757}

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

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

Once we have solved for \( c_2\) , we can use Equation \ref{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 \( c_1\ref{0} = 1\) and \( c_2\ref{0} = 0\) . It is easily demonstrated that the appropriate solutions are

\( c_2(t) =\) \ref{759} \( c_1(t)=\) \( \exp[\,{\rm i}\,(\omega-\omega_{21})\,t/2]\,\cos\left( [\gamma^2/\hbar^2+(\omega-\omega_{21})^{\,2}/4]^{1/2}\,t\right)\) \ref{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

\( P_2(t)\) \ref{761} \( P_1(t)\) \( = 1 - P_2(t).\) \ref{762}

Equation \ref{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

\( P_1(t)\) \( =\cos^2 (\gamma \,t / \hbar),\) \ref{763} \( P_2(t)\) \( = \sin^2 (\gamma \,t/\hbar ).\) \ref{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.

Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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