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.

$$\hat{A} f = k f$$ (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

$$\hat{A} \psi({\bf r}) = a \psi({\bf r})$$ (50)

then

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

$$= \int \psi^{*}({\bf r}) a \psi({\bf r}) d{\bf r}$$

$$= a \int \psi^{*}({\bf r}) \psi({\bf r}) d{\bf r}$$

$$= a$$

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

$$<A> = \frac{\int \psi^{*}({\bf r}) \hat{A} \psi({\bf r})} d{\bf r} {\int \psi^{*}({\bf r}) \psi({\bf r})} d{\bf r}$$ (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 $b$.

$$\psi = c_a \psi_a + c_b \psi_b$$ (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?

$$<A> = \int \psi^{*} \hat{A} \psi$$ (54)

$$= \int \left[ c_a \psi_a + c_b \psi_b \right]^{*} \hat{A} \left[ c_a \psi_a + c_b \psi_b \right]$$

$$= \int \left[ c_a \psi_a + c_b \psi_b \right]^{*} \left[ a c_a \psi_a + b c_b \psi_b \right]$$

$$= a \vert c_a\vert^2 \int \psi_a^{*} \psi_a + b c_a^{*} c_b \int \p... ... c_b^{*} c_a \int \psi_b^{*} \psi_a + b \vert c_b\vert^2 \int \psi_b^{*} \psi_b$$

$$= a \vert c_a\vert^2 + b \vert c_b\vert^2$$

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.