Mathematical Methods for Physics and Engineering - Matematica.NET

19.1 OPERATOR FORMALISMspectrum of the system is continuous. This system has discrete negative andcontinuous positive eigenvalues for the operator corresponding to the total energy(the Hamiltonian).◮Using the Dirac notation, show that the eigenvalues of an Hermitian operator are real.Let | a〉 be an eigenstate of Hermitian operator A corresponding to eigenvalue a, thenHence,⇒⇒A| a〉 = a| a〉,〈a|A| a〉 = 〈a|a| a〉 = a〈a| a〉,and〈a|A † = a ∗ 〈a|,〈a|A † | a〉 = a ∗ 〈a| a〉,〈a|A| a〉 = a ∗ 〈a| a〉, since A is Hermitian.(a − a ∗ )〈a| a〉 = 0,⇒ a = a ∗ , since 〈a| a〉 ≠0.Thus a is real. ◭It is not our intention to describe the complete axiomatic basis of quantummechanics, but rather to show what can be learned about linear operators, andin particular about their eigenvalues, without recourse to explicit wavefunctionson which the operators act.Before we proceed to do that, we close this subsection with a number of results,expressed in Dirac notation, that the reader should verify by inspection or byfollowing the lines of argument sketched in the statements. Where a sum overa complete set of eigenvalues is shown, it should be replaced by an integral forthose parts of the eigenvalue spectrum that are continuous. With the notationthat |a n 〉 is an eigenstate of Hermitian operator A with non-degenerate eigenvaluea n (or, if a n is k-fold degenerate, then a set of k mutually orthogonal eigenstateshas been constructed and the states relabelled), we have the following results.A| a n 〉 = a n | a n 〉,〈a m |a n 〉 = δ mn (orthonormality of eigenstates), (19.5)A(c n | a n 〉 + c m | a m 〉)=c n a n | a n 〉 + c m a m | a m 〉 (linearity). (19.6)The definitions of the sum and product of two operators are(A + B)| ψ〉 ≡A| ψ〉 + B| ψ〉, (19.7)AB| ψ〉 ≡A(B| ψ〉) (≠ BA| ψ〉 in general), (19.8)⇒ A p | a n 〉 = a p n| a n 〉. (19.9)651