Eigenfunctions and Eigenvalues

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Eigenfunctions and Eigenvalues

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.

$\begin{displaymath}\hat{A} f = k f \end{displaymath}$

(49)

where k is a constant called the eigenvalue. It is easy to show that if $\hat{A}$ is a linear operator with an eigenfunction g, then any multiple of g is also an eigenfunction of $\hat{A}$ .

When a system is in an eigenstate of observable A (i.e., when the wavefunction is an eigenfunction of the operator $\hat{A}$ ) then the expectation value of A is the eigenvalue of the wavefunction. Thus if

$\begin{displaymath}\hat{A} \psi({\bf r}) = a \psi({\bf r}) \end{displaymath}$

(50)

then

<A>	=	$\displaystyle \int \psi^{*}({\bf r}) \hat{A} \psi({\bf r})$	(51)
	=	$\displaystyle \int \psi^{*}({\bf r}) a \psi({\bf r})$
	=	$\displaystyle a \int \psi^{*}({\bf r}) \psi({\bf r})$
	=	a

assuming that the wavefunction is normalized to 1, as is generally the case. In the event that $\psi({\bf r})$ is not or cannot be normalized (free particle, etc.) then we may use the formula

$\begin{displaymath}<A> = \frac{\int \psi^{*}({\bf r}) \hat{A} \psi({\bf r})} {\int \psi^{*}({\bf r}) \psi({\bf r})} \end{displaymath}$

(52)

What if the wavefunction is a combination of eigenstates? Let us assume that we have a wavefunction which is a linear combination of two eigenstates of $\hat{A}$ with eigenvalues a and b.

$\begin{displaymath}\psi = c_a \psi_a + c_b \psi_b \end{displaymath}$

(53)

where $\hat{A} \psi_a = a \psi_a$ and $\hat{A} \psi_b = b \psi_b$ . Then what is the expectation value of A?

<A>	=	$\displaystyle \int \psi^{*} \hat{A} \psi$	(54)
	=	$\displaystyle \int \left[ c_a \psi_a + c_b \psi_b \right]^{*} \hat{A} \left[ c_a \psi_a + c_b \psi_b \right]$
	=	$\displaystyle \int \left[ c_a \psi_a + c_b \psi_b \right]^{*} \left[ a c_a \psi_a + b c_b \psi_b \right]$
	=	$\displaystyle a \vert c_a\vert^2 \int \psi_a^{} \psi_a + b c_a^{} c_b \int \p... ... c_b^{} c_a \int \psi_b^{} \psi_a + b \vert c_b\vert^2 \int \psi_b^{*} \psi_b$
	=	a \|c_a\|² + b \|c_b\|²

assuming that $\psi_a$ and $\psi_b$ are orthonormal (shortly we will show that eigenvectors of Hermitian operators are orthogonal). Thus 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.

Next: Hermitian Operators Up: Operators Previous: Linear Operators