Chapter 7. The Eigenvalue Problem

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7. The Eigenvalue Problem, December 17, 2009 2 aspects of the eigenvalue problem for matrices and do not discuss numerical methods for solving it. When we need a numerical solution, we use Mathematica. For demanding calculations, we must use Fortran, C, orC++, since Mathematica is less efficient. These computer languages have libraries of programs designed to give you the eigenvalues and the eigenvectors when you give them the matrix. § 2 Some general considerations. It turns out that the eigenvalue problem A|ψ = λ|ψ has many solutions and not all of them are physically meaningful. We are interested in this eigenvalue problem because, if it arises from an observable, the eigenvalues are the spectrum of the observable and some of the eigenfunctions are the pure states of the observable. In addition, the operator representing the observable is Hermitian, because its spectrum, which consists of all the values the observable can take, just contain only real numbers. Before we try to solve the eigenvalue problem, we need to decide how we recognize which eigenstates correspond to pure states: these are the only eigenstates of interest to us. In Chapter 2, we examined the definition of the probability and concluded that kets |a i and |a j representing pure states must satisfy the orthonormalization condition a i | a j = δ ij (4) § 3 Normalization. Let us assume that we are interested in solving the eigenvalue equation Â|ψ = λ|ψ (5) It is easy to see that if α is a complex number and |ψ satisfies Eq. 5 then |η = α|ψ satisfies the same equation. For every given eigenvalue we have as many eigenstates as complex numbers. The one that is a pure state must be normalized. We can therefore determine the value of α by requiring that |η satisfies η | η =1 (6) Using Eq. 5 in Eq. 6 (and the properties of the scalar product) leads to α ∗ αψ | ψ =1 (7) or α ∗ α = 1 ψ | ψ (8)
7. The Eigenvalue Problem, December 17, 2009 3 This gives α = e iφ ψ | ψ (9) where φ is a real number, which cannot be determined by solving Eq. 8. The ket e iφ |η = |ψ (10) ψ | ψ is normalized and therefore is a pure state of Â corresponding to the eigenvalue λ. If the system is in the state |η and we measure A, we are certain that the result is λ. In practice, most computer programs that calculate eigenstates do not normalize them. If the computer produces |ψ as the eigenket corresponding to eigenvalue λ, youmustuseEq.10tofind the eigenstate |η that is a pure state. Since e iφ in Eq. 10 is an irrelevant phase factor, we can drop it (i.e. take φ = 0); the pure state corresponding to λ is |η = |ψ ψ | ψ (11) § 4 Orthogonality. The pure states must also be orthogonal. Because an operator Â, representing an observable A, is always Hermitian, one can prove that eigenstates corresponding to different eigenvalues must be orthogonal. That is, if Â|η i = λ i |η i and Â|η j = λ j |η j (12) and then Theproofiseasy.Wehave λ i = λ j (13) η i | η j =0 (14) Âη i | η j −η i | Âη j =0 (15) because Â is Hermitian. Using Eqs. 12 in Eq. 15 gives (λ i − λ j )η i | η j =0
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Chapter 7. The Eigenvalue Problem

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