Chapter 7. The Eigenvalue Problem

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7. The Eigenvalue Problem, December 17, 2009 4 When λ i = λ j ,wemusthave η i | η j =0, which is what we wanted to prove. So, when you solve an eigenvalue equation for an observable, the eigenstates corresponding to different eigenvalues must be orthogonal. If they are not, there is something wrong with the program calculating the eigenvectors. What happens if some eigenvalues are equal to each other Physicists call such eigenvalues ‘degenerate’, mathematicians call them ‘multiple’. If |η i and |η j belong to the same eigenvalue (i.e. if Â|η i = λ|η i and Â|η j = λ|η j ) then |η i and |η j are under no obligation to be orthogonal to each other (only to, according to the fact in the previous §, all eigenvectors belonging to eigenvalues that differ from λ). Most numerical procedures for solving eigenvalue problems will provide non-orthogonal degenerate eigenvectors. These can be orthogonalized with the Gram-Schmidt procedure and then normalized. The resulting eigenvectors are now pure states. You will see how this is done in an example given later in this chapter. It is important to keep in mind this distinction between eigenstates and pure states. All pure states are eigenstates but not all eigenstates are pure states. § 5 The eigenvalue problem in matrix representation: a review. Let |a 1 , |a 2 , ..., |a n , ...be a complete orthonormal basis set. This set need not be a set of pure states of an observable; we can use any complete basis set that a mathematician can construct. Orthonormality means that Because the set is complete, we have a i | a j = δ ij for all i, j (16) Î N |a n a n | (17) n=1 Eq. 17 ignores the continuous spectrum and truncates the infinite sum in the completeness relation. We can act with each a m |, m =1, 2,...,N, on the eigenvalue equation to turn it into the following N equations: a m | Â | ψ = λa m | ψ, m =1, 2,...,N (18)
7. The Eigenvalue Problem, December 17, 2009 5 Now insert Î, asgivenbyEq.17,betweenÂ and |ψ to obtain N a m | Â | a na n | ψ = λa m | ψ, m =1, 2,...,N (19) n=1 The complex numbers a n | ψ are the coordinates of |ψ in the {|a n } N n=1 representation. If we know them, we can write |ψ as |ψ = N |a n a n | ψ (20) n=1 It is easy to rewrite Eq. 19 in a form familiar from linear matrix algebra. Let us denote ψ i ≡a i | ψ, i =1, 2,...,N (21) and A mn ≡a m | Â | a n (22) With this notation, Eq. 19 becomes and Eq. 20, N A mn ψ n = λψ m , m =1, 2,...,N (23) n=1 |ψ = N |a n ψ n (24) n=1 The sum in Eq. 23 is the rule by which the matrix A, having the elements A mn , acts on the vector ψ, having the coordinates ψ n . This equation is called the eigenvalue problem for the matrix A and it is often written as Aψ = λψ (25) (which resembles the operator equation Â|ψ = λ|ψ) oras ⎛ ⎞ ⎛ ⎞ ⎛ A 11 A 12 ··· A 1N ψ 1 ψ 1 A 21 A 22 ··· A 2N ψ 2 ⎜ ⎝ . . .. . ⎟ ⎜ ⎟ . ⎠ ⎝ . = λ ψ 2 ⎜ ⎠ ⎝ . A N1 A N2 ··· A NN ψ N ψ N ⎞ ⎟ ⎠ (26) Eqs. 25 and 26 are different ways of writing Eq. 23, which, in turn, is the representation of the equation Â|ψ = λ|ψ in the basis set {|a 1,...,|a N }.
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Chapter 7. The Eigenvalue Problem

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