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Lecture 5: Eigenvalue Equations and Operators The material in this lecture covers the following in Atkins. 11.5 The informtion of a wavefunction (b) eigenvalues and eigenfunctions (c) operators Lecture on-line Eigenvalue Equations and Operators (PDF) Eigenevalue Equations and Operators (PowerPoint) Handout for lecture 5 (PDF) Tutorials on-line Reminder of the postulates of quantum mechanics The postulates of quantum mechanics (This is the writeup for Dry-lab-II)( This lecture has covered postulate 4) Basic concepts of importance for the understanding of the postulates Observables are Operators - Postulates of Quantum Mechanics Expectation Values - More Postulates Forming Operators Hermitian Operators Dirac Notation Use of Matricies Basic math background Differential Equations Operator Algebra Eigenvalue Equations Extensive account of Operators Basic math background Differential Equations Operator Algebra Eigenvalue Equations Extensive account of Operators Audio-visuals on-line Postulates of Quantum mechanics (PDF) Postulates of Quantum mechanics (HTML) Postulates of quantum mechanics (PowerPoint ****) Slides from the text book (From the CD included in Atkins ,**) Quantum mechanical principles..Eigenfunctions The Schrödinger equation 2 2 (x) (x)V(x) E (x) 2m x 2 can be rewritten as 2 2 V(x) 2m x 2 (x) E(x) or : ˆ (x) E(x); H ˆ = H 2 2 V(x) 2m x 2 ˆ is the quantum mechanical Hamiltonian where H Quantum mechanical principles..Eigenfunctions ˆ E is an example The Schrödinger equation H of an eigenfunction equation (operator)(function) (cons tant )(samefunction) (operator)(eigenfunction) = (eigenvalue)(eigenfunction) Quantum mechanical principles..Eigenfunctions ˆ we have a function f(x) such that If for an operator A ˆ f(x) = kf(x) (where k is a constant) A ˆ than f(x) is said to be an Eigenfunction of A with the eigenvalue k e.g. d exp[2x ] 2 exp[2x ] dx d thus exp[2x] is an eigenfunction to dx with eigenvalue 2 Quantum mechanical principle.. Operators Aˆ f (x ) g(x ) : general definition of operator An operator is a rule that transforms a given function f into another function. We indicate an operator with a circumflex '^' also called 'hat'. ˆ Operator A d dx 3 cos() Function f ˆ f(x) A f f'(x) f x 3f cosx x x Quantum mechanical principle.. Operators Rules for operators: (Aˆ Bˆ )f (x ) Aˆ f (x ) Bˆ f (x ) : Sum of operators (Aˆ Bˆ )f (x ) Aˆ f (x ) Bˆ f (x ) : Dif. of operators d ˆ = Example D dx 3 3 3 ˆ ˆ ˆ (D 3)(x 5) D (x 5) 3(x 5) 3x (3x 15) 2 3 3x 3x 15 2 3 Quantum mechanical principle.. Operators Aˆ Bˆ f (x ) Aˆ [Bˆ f (x )] : product of operators ˆ' We first operate on f with the operator 'B on the right of the operator product, and ˆ f) and then take the resulting function (B ˆ on the left operate on it with the operator A of the operator product. d ˆ Example D = ; xˆ x dx Dˆ xˆf (x ) Dˆ (xf (x )) f (x ) xf '(x ) xˆDˆ f (x ) xˆ (Dˆ f (x )) xf '(x ) Quantum mechanical principle.. Operators Operators do not necessarily obey the commutative law: Aˆ Bˆ Bˆ Aˆ 0 : Aˆ Bˆ Bˆ Aˆ [ Aˆ , Bˆ ] 0 ˆ = X 2; B ˆ = Example : A Cummutator : d dx 2f) df d(x df 2 2 ˆ ˆ ˆ ˆ ABf x : BAf = 2xf x dx dx dx ˆ ,B ˆ ]f 2xf [A Quantum mechanical principle.. Operators The square of an operator is defined as the product of ˆ2 = A ˆA ˆ the operator with itself: A ˆ= d Examples : D dx ˆD ˆ f(x) = D ˆ (Dˆ f(x)) = D ˆ f'(x) f "(x ) D d2 D dx 2 ˆ2 Quantum mechanical principle.. Operators ˆ ,B ˆ ,C ˆ , etc. We shall be dealing with linear operatorsA where the follow rules apply Aˆ {f (x ) g(x )} Aˆ f (x ) Aˆ g(x) Aˆ {kf (x )} kAˆ f (x ) Some linear operators: d d2 x;x ; ; 2 dx dx 2 Multiplicative Some non - linear operators: Differential cos; : 2 For linear operators the following identities apply: ˆ +B ˆ )C ˆ =A ˆC ˆ +B ˆC ˜; A ˆ (B ˆ +C ˆ) =A ˆB ˆ +A ˆC ˆ (A Quantum mechanical principles..Eigenfunctions ˆ be a linear operator * Let A with the eigenfunction f and the eigenvalue k** . Demonstrate that cf also is an eigenfunction to with the same eigenvalue k if c is a constant Must show Aˆ (cf ) k (cf ) proof : ˆ (cf ) cA ˆf A ˆ A ckf k(cf ) ˆ is a Linear operator A Rearrangement of constant factors and QED ˆ f is an eigenfunction of A *A ˆ (cf ) cAˆ f c is a constant d f is a function e.g. A = dx ** A ˆ f kf Quantum mechanical principle.. Operators General Commutation Relations The following relations are readily shown ^ ^ ,B ^ ^ n ,A ] [ A [ A ^ [k A ^ [ A ^ ^ ^ ] = - [ B ,A ^ ,B = 0 n=1,2,3,....... ^ ]=[ A ^ ^ , B +C ^ ] ^ ^ ,k B ^ ^ ,B ]=[ A ^ ^ ^ ,B ] = k[ A ^ ]+[ A ^ ^ ,C ^ ] ] ^ [ A +B ,C ] = [ A ,B ] + [ A ,C ] Quantum mechanical principle.. Operators ^ [ A ^^ ,B C ^ ^ ^ ^ ^ ^ ^ , C ] = [ A , B ]C ^ [ A B , C ]=[ A The operators are differential or ^ A ^ , B ^ ^ ^ + B [A , C ] ^ ] B ^ ^ + A [ B ^ , C multiplicative operators ^ , C ] Quantum mechanical principles..Eigenfunctions ˆ will have a set of A linear operator A eigenfunctions fn (x ) {n = 1,2,3..etc} and associated eigenvalues kn such that : ˆ fn (x ) k n fn (x ) A The set of eigenfunction {fn (x ),n 1..} is orthonormal : fi (x )fj (x )dx ij all space o if i j 1 if i= j Quantum mechanical principles..Eigenfunctions Examples of operators and their eigenfunctions Example 1 2 3 4 Operator Eigenfunction Eigenvalue x 2 x 2 x 2 x exp[ikx ] ik exp[ikx ] k 2 coskx k 2 sinkx k 2 What you should learn from this lecture 1. In an eigenvalue equation : ; an operator works on a function to give the function back times a constant . The function is called an eigenfunction and the constant . ˆ ) is a rule that transforms a given 2. An operator ( A ˆ f = g. function f into another function g as A We indicate an operator with a circumflex '^' also called 'hat'. 3. Oprators obays : (Aˆ Bˆ )f (x ) Aˆ f (x ) Bˆ f (x ) : Sum of operators (Aˆ Bˆ )f (x ) Aˆ f (x ) Bˆ f (x ) : Dif. of operators Aˆ Bˆ f (x ) Aˆ [Bˆ f (x )] : product of operators ˆ (B ˆC ˆ )f(x) = (A ˆB ˆ )C ˆ f(x): associative law of multiplication A ˆB ˆ B ˆA ˆ = [A ˆ ,B ˆ ] 0; Operators do not commute, A ˆ ,B ˆ ] is call the commutator. order of operators matters. [A What you should learn from this lecture 4. Linear operators obey : Aˆ {f (x ) g(x )} Aˆ f (x ) Aˆ g(x) Aˆ {kf (x )} kAˆ f (x ) 2 d d Some linear operators are : x;x 2; ; dx dx 2 ˆ will have a set of eigenfunctions 5. A linear operator A fn (x) {n = 1,2,3..etc} and associated eigenvalues kn ˆ fn (x) kn fn (x) such that : A The set of eigenfunction {fn (x),n 1..} is orthonormal : * fi (x)(fj (x)) dx ij all space