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21.2: Eigenfunctions and Eigenvalues

  • Page ID
    416093
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    As we have already seen, an eigenfunction of an operator \(\hat{A}\) is a function \(f\) such that the application of \(\hat{A}\) on \(f\) gives \(f\) again, times a constant:

    \[ \hat{A} f = k f, \label{22.2.1} \]

    where \(k\) is a constant called the eigenvalue.

    When a system is in an eigenstate of observable \(A\) (i.e., when the wave function is an eigenfunction of the operator \(\hat{A}\)) then the expectation value of \(A\) is the eigenvalue of the wave function. Therefore:

    \[ \hat{A} \psi({\bf r}) = a \psi({\bf r}), \label{22.2.2} \]

    then:

    \[\begin{equation} \begin{aligned}\langle A \rangle &= \int \psi^{*}({\bf r}) \hat{A} \psi({\bf r}) d{\bf r} \\ &= \int \psi^{*}({\bf r}) a \psi({\bf r}) d{\bf r} \\ &= a \int \psi^{*}({\bf r}) \psi({\bf r}) d{\bf r} = a, \end{aligned} \end{equation} \label{22.2.3} \]

    which implies that:

    \[ \int \psi^{*}({\bf r}) \psi({\bf r}) d{\bf r} = 1. \label{22.2.4} \]

    This property of wave functions is called normalization, and in the one-electron TISEq guarantees that the maximum probability of finding an electron over the entire space is one.1

    A unique property of quantum mechanics is that a wave function can be expressed not just as a simple eigenfunction, but also as a combination of several of them. We have in part already encountered such property in the previous chapter, where complex hydrogen orbitals have been combined to form corresponding linear ones. As a general example, let us consider a wave function written as a linear combination of two eigenstates of \(\hat{A}\), with eigenvalues \(a\) and \(b\):

    \[ \psi = c_a \psi_a + c_b \psi_b, \label{22.2.5} \]

    where \(\hat{A} \psi_a = a \psi_a\) and \(\hat{A} \psi_b = b \psi_b\). Then, since \(\psi_a\) and \(\psi_b\) are orthogonal and normalized (usually abbreviated as orthonormal), the expectation value of \(A\) is:

    \[\begin{equation} \begin{aligned}\langle A \rangle &= \int \psi^{*} \hat{A} \psi d{\bf r} \\ &= \int \left[ c_a \psi_a + c_b \psi_b \right]^{*} \hat{A} \left[ c_a \psi_a + c_b \psi_b \right] d{\bf r}\\ &= \int \left[ c_a \psi_a + c_b \psi_b \right]^{*} \left[ a c_a \psi_a + b c_b \psi_b \right] d{\bf r}\\ &= a \vert c_a\vert^2 \int \psi_a^{*} \psi_a d{\bf r} + b c_a^{*} c_b \int \psi_a^{*} \psi_b d{\bf r} + a c_b^{*} c_a \int \psi_b^{*} \psi_a d{\bf r} + b \vert c_b\vert^2 \int \psi_b^{*} \psi_b d{\bf r}\\ &= a \vert c_a\vert^2 + b \vert c_b\vert^2. \end{aligned} \end{equation} \label{22.2.6} \]

    This result shows that the average value of \(A\) is a weighted average of eigenvalues, with the weights being the squares of the coefficients of the eigenvectors in the overall wavefunction.2


    1. ︎Imposing the normalization condition is the best way to find the constant \(A\) in the solution of the TISEq for the particle in a box, a topic that we delayed in chapter 20.︎
    2. This section was adapted in part from Prof. C. David Sherrill’s A Brief Review of Elementary Quantum Chemistry Notes available here.

    This page titled 21.2: Eigenfunctions and Eigenvalues is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Roberto Peverati.

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