Download Eigenvalue Problem & Operator Functions in Quantum Mechanics (PHYS 560, NIU, Fall 2007) and more Study notes Quantum Physics in PDF only on Docsity! Handout 2 from PHYS 560 Northern Illinois University Fall 2007 (Dated: version printed August 21, 2007) Brief notes on concepts and topics from Chapter 1 Sections 8 and 9. Shankar’s Chapter One: Mathematical Introduction 1.8. THE EIGENVALUE PROBLEM • Review that linear operator acts on arbitrary non-zero ket to change it to another non-zero ket in the same vector space. Notation, Ω|V 〉 = |V ′〉. same vector space means that we can expand |V 〉 or |V ′〉 using the same basis set. Convince yourself that this explicit or implict assumption in proofs. The eigenvalue (ω) is just a scalar. |ω〉 is a ket, (an object in the vector space). Ω is the operator, can be expressed as a matrix. • Eigenvalues and eigenkets are found for a particular operator. Different operator may mean totally different values and kets. • Eigenvalue equation in notation: Ω|V 〉 = ω|V 〉 or more commonly seen the shorthand Ω|ω〉 = ω|ω〉. review and understand that the ω symbol above is doing double duty. When inside the ket, it denotes the ket associated with the eigenvalue. Occasionally, because is just a scalar value, the eigenvalue ω might get put anywhere convenient and might appear inside the ket (example |αV 〉 = α|V 〉 or 〈αV | = α∗〈V |.). But this is rarely done with the eigenvalue because of the possibility of confusion. • If [Ω̂, Λ̂] = 0 then Ω̂|V 〉 = ω|V 〉 and Λ̂|V 〉 = λ|V 〉. But there is no reason to assume that λ = ω. Equality of eigenkets between two operators doesn’t imply a relationship between the eigenvalues associated with the kets. • Characteristic equation and solution to the eigenvalue equation convince yourself that you understand the argument leading to the eigenvalue equation as expressed by det(Ω− ωI) = 0. • Obviously, be able to find the determinant of an arbitrary n×n matrix and thus be able to find the eigenvalues of an operator in n dimensional space. Be able to evaluate the the eigenvectors associated with those eigenvalues. • Operators with special properties: Hermitian and Unitary. (In QM we will usually always use Hermitian operators so that the eigenvalues are ’real’ scalar and not complex. More on this later in course). Adjoint (or sometimes called Hermitian Adjoint) is defined for pretty much any operator. It is just (in matrix notation) Ω†ij = Ω ∗ ji. However, when Ω † ij = Ωij , then the operator is ‘Hermitian’. • a variety of useful relationships are true for Hermitan and Unitary operators det Ω = ∏n i−1 ωi and Tr Ω = ∑n i=1 ωi. Check out if chapter has other goodies here. • Follow proof that eigenvalues of Hermitian operators are real scalars. What assumptions are needed to go backwards? (i.e., if eigenvalues are real, is the operator Hermitian?) • Degeneracy: is just when more than one eigenvalues are the same and be able to calculate eigenkets. convince yourself that you understand why this gives more freedom in the choice of eigenvectors. • Relation of rotation to diagonalization of operators. • Convince yourself that a matrix composed of the eigenkets of a Hermitian operator will be a unitary operator. (And can be used to diagonalize the Hermitian operator into a simple representationω1 0 0 · · ·0 ω2 0 · · ·... 0 ω3 · · · .
