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7.2: Two-State System

  • Page ID
    1217
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    Let us start by considering time-independent perturbation theory, in which the modification to the Hamiltonian, \( H_1\) , has no explicit dependence on time. It is usually assumed that the unperturbed Hamiltonian, \( H_0\) , is also time-independent.

    Consider the simplest non-trivial system, in which there are only two independent eigenkets of the unperturbed Hamiltonian. These are denoted

    \( H_0 \,\vert 1\rangle\) \( =\) \( E_1 \,\vert 1\rangle,\) \ref{584} \( H_0 \,\vert 2\rangle\) \( =\) \( E_2 \,\vert 2\rangle.\) \ref{585}


    It is assumed that these states, and their associated eigenvalues, are known. Because \( H_0\) is, by definition, an Hermitian operator, its two eigenkets are mutually orthogonal and form a complete set. The lengths of these eigenkets are both normalized to unity. Let us now try to solve the modified energy eigenvalue problem

    \( (H_0 + H_1) \,\vert E\rangle = E\,\vert E\rangle.\) \ref{586}

    In fact, we can solve this problem exactly. Since the eigenkets of \( H_0\) form a complete set, we can write

    \( \vert E\rangle = \langle 1\vert E\rangle \vert 1\rangle + \langle 2\vert E\rangle \vert 2\rangle.\) \ref{587}

    Right-multiplication of Equation \ref{586} by \( \langle 1\vert\) and \( \langle 2\vert\) yields two coupled equations, which can be written in matrix form:

    $ \left( \begin{array}{c c} E_1 -E + e_{11} & e_{12} \\ e_{12}^{\,\...
...ngle\end{array} \!\right)= \left(\!\begin{array}{c}0\\ 0 \end{array}\! \right).$ \ref{588}

    Here,

    \( e_{11}\) \( =\) \( \langle 1\vert\,H_1\, \vert 1\rangle,\) \ref{589} \( e_{22}\) \( =\) \( \langle 2 \vert\,H_1\, \vert 2\rangle,\) \ref{590} \( e_{12}\) \( =\) \( \langle 1\vert\,H_1\,\vert 2\rangle.\) \ref{591}


    In the special (but common) case of a perturbing Hamiltonian whose diagonal matrix elements (in the unperturbed eigenstates) are zero, so that

    \( e_{11} = e_{22} = 0,\) \ref{592}

    the solution of Equation \ref{588} (obtained by setting the determinant of the matrix equal to zero) is

    \( E = \frac{(E_1+E_2) \pm \sqrt{(E_1-E_2)^{\,2} + 4\,\vert e_{12}\vert^{\,2}}}{2}.\) \ref{593}

    Let us expand in the supposedly small parameter

    \( \epsilon = \frac{\vert e_{12}\vert}{\vert E_1-E_2\vert}.\) \ref{594}

    We obtain

    \( E\simeq \frac{1}{2} \,(E_1+E_2) \pm \frac{1}{2}\,(E_1-E_2)\,(1+2\,\epsilon^2 + \cdots).\) \ref{595}

    The above expression yields the modifications to the energy eigenvalues due to the perturbing Hamiltonian:

    \( E_1'\) \( =\) \( E_1 + \frac{\vert e_{12}\vert^{\,2}}{E_1-E_2} + \cdots,\) \ref{596} \( E_2'\) \( =\) \( E_2 - \frac{\vert e_{12}\vert^{\,2}}{E_1-E_2} + \cdots.\) \ref{597}


    Note that \( H_1\) causes the upper eigenvalue to rise, and the lower eigenvalue to fall. It is easily demonstrated that the modified eigenkets take the form

    \( \vert 1\rangle'\) \( =\) \( \vert 1\rangle + \frac{e_{12}^{~\ast}}{E_1-E_2}\, \vert 2\rangle + \cdots,\) \ref{598} \( \vert 2\rangle'\) \( =\) \( \vert 2\rangle - \frac{e_{12}}{E_1-E_2}\, \vert 1\rangle +\cdots.\) \ref{599}


    Thus, the modified energy eigenstates consist of one of the unperturbed eigenstates with a slight admixture of the other. Note that the series expansion in Equation \ref{595} only converges if \( 2\,\vert\epsilon\vert<1\) . This suggests that the condition for the validity of the perturbation expansion is

    \( \vert e_{12}\vert < \frac{\vert E_1-E_2\vert}{2}.\) \ref{600}

    In other words, when we say that \( H_1\) needs to be small compared to \( H_0\) , what we really mean is that the above inequality needs to be satisfied.

    Contributors

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

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)

    This page titled 7.2: Two-State System is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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