21.2: Eigenfunctions and Eigenvalues
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
- ︎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 .︎
- This section was adapted in part from Prof. C. David Sherrill’s A Brief Review of Elementary Quantum Chemistry Notes available here .