Two-Level Systems

There are many problems in Quantum Chemistry that can be modeled in terms of the two-level Hamiltonian (i.e., a state-space with only two dimensions). Examples include electron transfer, proton transfer, and isomerization reactions.

Consider two states $\vert\phi_1>$ and $\vert\phi_2>$ , of a system. Assume that these states have similar energies, and , both of them well separated from all of the other energy levels of the system,

$\displaystyle \hat{H}_0\vert \phi_1> = E_1 \vert\phi_1>,$

$\displaystyle \hat{H}_0\vert \phi_2> = E_2 \vert\phi_2>.$

In the presence of a perturbation,

$\displaystyle W = \left ( \begin{array}{cc} 0 & \Delta \\ \Delta & 0 \end{array} \right ),$

the total Hamiltonian becomes

. Therefore, states $\vert\phi_1>$ and $\vert\phi_2>$ are no longer eigenstates of the system.

The goal of this section is to compute the eigenstates of the system in the presence of the perturbation W. The eigenvalue problem,

$\displaystyle \left ( \begin{array}{cc} H_{11} & H_{12} \\ H_{21} & H_{22} \end... ...\end{array} \right ) \left ( \begin{array}{c} C_1 \\ C_2 \end{array} \right ),$

is solved by finding the roots of the characteristic equation, $(H_{11}-E)(H_{22}-E)- H_{12}H_{21} =0$ . The values of

that satisfy such equation are,

$\displaystyle E_{\pm} = \frac{(E_1+E_2)}{2} \pm \sqrt{\left(\frac{E_1-E_2}{2}\right)^2 + \Delta^2}.$

These eigenvalues $E_{\pm}$ can be represented as a function of the energy difference , according to the following diagram:

$\begin{picture}(50,40)(-10,10) \linethickness{1pt}\thinlines\par\put(20,0){\vect... ...{$0$}} % put(12.5,10)\{ circle\{80\}\} % put(0,0)\{ circle*\{1\}\} \end{picture}$

Note that and cross each other, but and repel each other. Having found the eigenvalues $E_{\pm}$ , we can obtain the eigenstates $\vert\psi_{\pm}> = C_{1\pm} \vert\phi_1> + C_{2\pm} \vert\phi_2>$ by solving for $C_{1\pm}$ and $C_{2\pm}$ from the following equations:

$C_{1\pm}(H_{11} -E_{\pm}) +C_{2\pm}H_{12} =0$ ,

$\sum_{j=1}^2C_{j\pm}^*C_{j\pm} =1.$

We see that in the presence of the perturbation the minimum energy state $\vert\psi_->$ is always more stable than the minimum energy state of the unperturbed system.

Example 1. Resonance Structure

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The coupling between the two states makes the linear combination of the two more stable than the minimum energy state of the unperturbed system.

Example 2. Chemical Bond

$\begin{picture}(50,40)(-10,10) \linethickness{1pt}\thinlines % put(20,0)\{ vecto... ...$H^+$}} \put(70,35){\makebox(0,0)[t]{$\leftarrow \vert \phi_1 >$}} \end{picture}$
The state of the system that involves a linear combination of these two states is more stable than because $<\phi_1 \vert H\vert \phi_2> \neq 0$ .

Time Evolution

Consider a two level system described by the Hamiltonian , with $H_0 \mid \phi_1 > = E_1 \mid \phi_1 >$ . Assume that the system is initially prepared in state $\mid \psi(0) > = \mid \phi_1 >$ . Due to the presence of the perturbation , state $\mid \phi_1 >$ is not a stationary state. Therefore, the initial state evolves in time according to the time-dependent Schrödinger Equation,

$\displaystyle i \hbar \frac{\partial \vert \psi >}{\partial t} = \left ( H_0 + W \right ) \vert\psi >,$

and becomes a linear superposition of states $\vert\phi_1>$ and $\vert\phi_2>$ ,

$\displaystyle \vert\psi(t)> = C_1(t) \vert\phi_1> + C_2(t) \vert\phi_2>.$

State $\mid \psi(t) >$ can be expanded in terms of the eigenstates $\vert\psi_{\pm}>$ as follows,

$\displaystyle \vert\psi(t)> = C_+(t) \vert\psi_+> + C_-(t) \vert\psi_->,$

where the expansion coefficients $C_{\pm}(t)$ evolve in time according to the following equations,

$\displaystyle i \hbar \frac{\partial C_+(t)}{\partial t} =E_+ C_+ (t),$

$\displaystyle i \hbar \frac{\partial C_-(t)}{\partial t} =E_- C_- (t).$

Therefore, state $\vert \psi(t) >$ can be written in terms of $\vert\psi_{\pm}>$ as follows,

$\displaystyle \vert \psi (t) > = C_+(0) e^{-\frac{i}{\hbar}E_+t}\vert\psi_+> +C_-(0) e^{-\frac{i}{\hbar}E_-t}\vert\psi_->.$

The probability amplitude of finding the system in state $\vert\phi_2>$ at time is,

$\displaystyle P_{12}(t) = \vert<\phi_2\vert\psi(t)>\vert^2 = C_2(t)^*C_2(t),$

which can also be written as follows,

$\displaystyle P_{12}(t) = \vert C_{2+} C_+(0)\vert^2 + \vert C_{2-} C_-(0)\vert^2 + C_{2+}^* C_+^*(0) C_{2-} C_-(0) e^{-\frac{i}{\hbar}(E_--E_+)t},$

where $C_{2\pm}=< \phi_2 \mid \Psi_{\pm}>$ . The following diagram represents $P_{12}(t)$ as a function of time:

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The frequency $\nu = (E_+ - E_-)/(\pi \hbar)$ is called Rabi Frequency. It is observed, e.g., in the absorption spectrum of (see Example 2). It corresponds to the frequency of the oscillating dipole moment which fluctuates according to the electronic configurations of $\vert\phi_1>$ and $\vert\phi_2>$ , respectively. The oscillating dipole moment exchanges energy with an external electromagnetic field of its own characteristic frequency and, therefore, it is observed in the absorption spectrum of the system.