    Next: Hermitian Operators Up: Operators Previous: Linear Operators

### Eigenfunctions and Eigenvalues

An eigenfunction of an operator is a function f such that the application of on f gives f again, times a constant. (49)

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

When a system is in an eigenstate of observable A (i.e., when the wavefunction is an eigenfunction of the operator ) then the expectation value of A is the eigenvalue of the wavefunction. Thus if (50)

then
 = (51) = = = a

assuming that the wavefunction is normalized to 1, as is generally the case. In the event that is not or cannot be normalized (free particle, etc.) then we may use the formula (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 with eigenvalues a and b. (53)

where and . Then what is the expectation value of A?
 = (54) = = = = a |ca|2 + b |cb|2

assuming that and 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