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Lecture 6: Operators and Quantum Mechanics The material in this lecture covers the following in Atkins. 11.5 The informtion of a wavefunction (c) Operators Lecture on-line Operators in quantum mechanics (PDF) Operators in quantum mechanics (HTML) Operators in Quantum mechanics (PowerPoint) Handout (PDF) Assigned Questions 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 3) 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 Historic development of quantum mechanics from classical mechanics The Development of Classical Mechanics Experimental Background for Quantum mecahnics Early Development of Quantum mechanics Audio-visuals on-line Early Development of Quantum mechanics Audio-visuals on-line Postulates of Quantum mechanics (PDF) (simplified version from Wilson) Postulates of Quantum mechanics (HTML) (simplified version from Wilson) Postulates of Quantum mechanics (PowerPoint ****) (simplified version from Wilson) Slides from the text book (From the CD included in Atkins ,**) Operators and Quantum Mechanics We now have Re view (Ia) A Quantum mechanical system is specified by the statefunction (x) (Ib) The state function (x) contains all information about the system we can know (Ic) A system described by the state function H(x) = E(x) has exactly the energy E Operators and Quantum Mechanics We have seen that a 'free' particle moving Re view in one dimension in a constant (zero) potential has the Hamiltonian 2 2 X Hˆ O The Schrodinger equation is 2 2 (x) E (x) 2m x 2 with the general solution : (x) Aexp ikx 2 2 k and energies E = 2m Bexp ikx 2m x 2 Operators and Quantum Mechanics How does the state function (x, t) give us information about an observable other than the energy such as the position or the momentum ? Good question Any observable ' ' can be expressed in classical physics in terms of x,y, z and px ,p y ,p z . Examples : = x, px , v x , p2 x , T, V(x), E Operators and Quantum Mechanics We can construct the corresponding operator from the substitution: Classical Mechanics Quantum Mechanics x px y py z pz ˆ (x,y,z, as xˆ x ; pˆ x i x yˆ y ; pˆ y i y zˆ z ; pˆ z i z d d d , , ) i dx i dy i dz Such as : ˆ ˆ ˆ xˆ , pˆ x , vˆx , pˆ 2 = x , T, V(x), E Re view Operators and Quantum Mechanics Im portan t news For an observable with the corresponding ˆ we have the eigenvalue equation : operator n nn (IIIa). The meassurement of the quantity represented by has as the o n l y outcome one of the values n n = 1, 2, 3 .... (IIIb). If the system is in a state described by n a meassurement of will result in the value n Quantum mechanical principle.. Operators ˆ For any such operator Im portan t news we can solve the eigenvalue problem ˆ n n n We obtain eigenfunctions and eigenvalues The only possible values that can arise from measurements of the physical observable are the eigenvalues n Postulate 3 Operators and Quantum Mechanics Im portan t news The x - component 'px ' of the linear momentum p pxex pye y pz ez Is represented by the operator pˆ x With the eigenfunctions Exp[ikx] ix and eigenvalue k Exp[ikx] = kExp[ikx] i x We note that k can take any value > k > Operators and Quantum Mechanics (x) Aexp ikx For A = 0 Bexp ikx New insight 2 2 k and energies E = 2m ikx (x ) B exp ˆx this wavefunction is also an eigenfunction to p With eigenvalue for pˆ x of - k 2 2 k Thus - (x ) describes a particle of energy E= 2m Px2 and momentum p x k ; note E = as it must be. 2m This system corresponds to a particle moving with constant velocity p We know nothing about its position vx x - k/m m since | (x) |2 B Operators and Quantum Mechanics (x) Aexp ikx For B = 0 New insight ikx and energies E = Bexp 2 2 k 2m (x ) A exp ikx ˆx this wavefunction is also an eigenfunction to p With eigenvalue for pˆ x of k Thus (x ) describes a particle of energy E= 2 2 k 2m Px2 and momentum p x k ; note E = as it must be. 2m This system corresponds to a particle moving with constant velocity px We know nothing about its position vx k/m m since | (x) |2 B Operators and Quantum Mechanics What about : (x ) A exp ikx d d ikx ikx ˆ p x (x) = A exp B exp i dx i dx A k expikx B k expikx How can we find px in this case ? ikx B exp ˆ x since: It is not an eigenfunction to p New insight ? 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 : * f (x)dx f (x) i j ij all space o if i j 1 if i= j Quantum mechanical principles..Eigenfunctions An example of an orthonormal set is the Cartesian unit vectors ei ei e j ij ei ei An example of an orthonormal function set is n (x) = 1 nx sin L L n = 1, 2, 3,4, 5.... L * n (x) m (x) nm o Quantum mechanical principles..Eigenfunctions The set of eigenfunction {fn (x ),n 1..} forms a complete set. That is, any function g(x) that depends on the same variables as the eigenfunctions can be written ei ; i = 1, 2,3 form a complete set all g(x) = anfn (x ) ei i=1 ei where an ei * fn (x) g(x)dx all space For any vector v v (v e1 )e1 (v e 2 )e2 (v e 3 )e3 Quantum mechanical principles..Eigenfunctions all In the expansion : g(x) = aifi (x ) (1) i=1 we can show that : an fn (x)* g(x)dx V * from the orthonormality : fi (x) fj (x)dx ij V A multiplication by fn (x) on both sides followed by integration affords all * all g(x) = aifi (x) fn (x) g(x)dx = ai fn (x)* fi (x)dx i=1 or : : aann or V * g(x)f (x )dx g(x)f (x) dx nn all space space all i=1 V ij Operators and Quantum Mechanics (x) Aexpikx Bexp ikx is a linear combination of two eigenfunctions to pˆ x px k How can we find px in this case ? px k What you should learn from this lecture 1. Postulate 3 ˆ For an observable with the corresponding operator we have the eigenvalue equation : n n n (i) The meassurement of the quantity represented by has as the o n l y outcome one of the values n n = 1, 2, 3 .... (ii) If the system is in a state described by n a meassurement of will result in the value n Illustrations : (x) A expikx is an eigenfunction to pˆ x with eigenvalue k (x) A expikx is an eigenfunction to pˆ x with eigenvalue - k Both are eigenfunctions to the Hamiltonian for a free particle 2 (ˆp )2 2 k2 x H= with eigenvalues E = 2m 2m (x) represents a free particle of momentum k (x) represents a free particle of momentum - k What you should learn from this lecture 2. Postulate 4. The set of eigenfunction {fn (x),n 1..} forms a complete set. That is, any function g(x) that depends on the same variables as the eigenfunctions can be written : all g(x) = anfn (x) where i=1 an g(x)fn (x)dx all space