Mathematical Methods for Physics and Engineering - Matematica.NET

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONSwhich is a statement of the orthogonality of y i and y j .If one (or more) of the eigenvalues is degenerate, however, we have differenteigenfunctions corresponding to the same eigenvalue, and the proof of orthogonalityis not so straightforward. Nevertheless, an orthogonal set of eigenfunctionsmay be constructed using the Gram–Schmidt orthogonalisation method mentionedearlier in this chapter and used in chapter 8 to construct a set of orthogonaleigenvectors of an Hermitian matrix. We repeat the analysis here for completeness.Suppose, for the sake of our proof, that λ 0 is k-fold degenerate, i.e.Ly i = λ 0 ρy i for i =0, 1,...,k− 1, (17.25)but that λ 0 is different from any of λ k , λ k+1 , etc. Then any linear combination ofthese y i is also an eigenfunction with eigenvalue λ 0 since∑k−1∑k−1∑k−1Lz ≡ L c i y i = c i Ly i = c i λ 0 ρy i = λ 0 ρz. (17.26)i=0i=0If the y i defined in (17.25) are not already mutually orthogonal then considerthe new eigenfunctions z i constructed by the following procedure, in which eachof the new functions z i is to be normalised, to give ẑ i , before proceeding to theconstruction of the next one (the normalisation can be carried out by dividingthe eigenfunction z i by ( ∫ ba z∗ i z iρdx) 1/2 ):z 0 = y 0 ,∫ b)z 1 = y 1 −(ẑ 0 ẑ0y ∗ 1 ρdx ,a∫ b) ∫ b)z 2 = y 2 −(ẑ 1 ẑ1y ∗ 2 ρdx −(ẑ 0 ẑ0y ∗ 2 ρdx ,aa.∫ b)∫ b)z k−1 = y k−1 −(ẑ k−2 ẑk−2y ∗ k−1 ρdx − ···−(ẑ 0 ẑ0y ∗ k−1 ρdx .aaEach of the integrals is just a number and thus each new function z i is, as can beshown from (17.26), an eigenvector of L with eigenvalue λ 0 . It is straightforwardto check that each z i is orthogonal to all its predecessors. Thus, by this explicitconstruction we have shown that an orthogonal set of eigenfunctions of anHermitian operator L can be obtained. Clearly the orthogonal set obtained, z i ,isnot unique.In general, since L is linear, the normalisation of its eigenfunctions y i (x) isarbitrary. It is often convenient, however, to work in terms of the normalisedeigenfunctions ŷ i (x), so that ∫ ba ŷ∗ i ŷiρdx= 1. These therefore form an orthonormal562i=0