Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Dirac bracket wikipedia , lookup
Coherent states wikipedia , lookup
Quantum state wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Density matrix wikipedia , lookup
Bra–ket notation wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Self-adjoint operator wikipedia , lookup
Commutators and the Correspondence Principle Formal Connection Q.M. Classical Mechanics Correspondence between Classical Poisson bracket of functions f ( x , p) and g( x , p) And Q.M. Commutator of operators f and g . Copyright – Michael D. Fayer, 2007 Commutator of Linear Operators A, B A B B A (This implies operating on an arbitrary ket.) If A and B numbers, = 0 Operators don’t necessarily commute. A B C A B C AQ Z B A C B A C B S T In General Z T A and B do not commute. Copyright – Michael D. Fayer, 2007 Classical Poisson Bracket f , g f g g f x p x p f f ( x , p) g g( x , p) These are functions representing classical dynamical variables not operators. Consider position and momentum, classical. x and p Poisson Bracket x p x p x , p x p p x x, p 1 zero Copyright – Michael D. Fayer, 2007 Dirac’s Quantum Condition "The quantum-mechanical operators f and g, which in quantum theory replace the classically defined functions f and g, must always be such that the commutator of f and g corresponds to the Poisson bracket of f and g according to i f , g f , g ." Dirac Copyright – Michael D. Fayer, 2007 This means i f ( x , p), g( x , p) f , g Poisson bracket of classical functions (commutator operates on a ket) Commutator of quantum operators Q.M. commutator of x and p. x , p i commutator Therefore, x , p Poisson bracket x, p 1 x , p i Remember, the relation implies operating on an arbitrary ket. This means that if you select operators for x and p such that they obey this relation, they are acceptable operators. The particular choice a representation of Q.M. Copyright – Michael D. Fayer, 2007 Schrödinger Representation p P i x momentum operator, –i times derivative with respect to x xx x position operator, simply x Operate commutator on arbitrary ket x, P S ( xP P x ) S x i x S S i x x Using the product rule i x S S x S x x i s . Therefore, x, P S i S and x, P i because the two sides have the same result when operating on an arbitrary ket. S Copyright – Michael D. Fayer, 2007 Another set of operators – Momentum Representation xxi p position operator, i times derivative with respect to p p p momentum operator, simply p A different set of operators, a different representation. In Momentum Representation, solve position eigenvalue problem. Get x , states of definite position. They are waves in p space. All values of momentum. Copyright – Michael D. Fayer, 2007 Commutators and Simultaneous Eigenvectors A S S B S S S are simultaneous Eigenvectors of operators A and B with egenvalues and . Eigenvalues of linear operators observables. A and B are different operators that represent different observables, e. g., energy and angular momentum. If S are simultaneous eigenvectors of two or more linear operators representing observables, then these observables can be simultaneously measured. Copyright – Michael D. Fayer, 2007 A S S B S S B A S B S AB S A S B S A S S S Therefore, Rearranging AB S B A S A B B A A B B A is the commutator of S 0 A and B, and since in general S 0, A, B 0 The operators A and B commute. Operators having simultaneous eigenvectors commute. The eigenvectors of commuting operators can always be constructed in such a way that they are simultaneous eigenvectors. Copyright – Michael D. Fayer, 2007 There are always enough Commuting Operators (observables) to completely define a system. Example: Energy operator, H, may give degenerate states. H atom 2s and 2p states have same energy. J 2 square of angular momentum operator j 1 for p orbital j 0 for s orbital But px, py, pz J z angular momentum projection operator 2 H , J , J z all commute. Copyright – Michael D. Fayer, 2007 Commutator Rules A , B B , A A , BC A , B C B A , C AB , C A , C B A B , C A , B , C B , C , A C , A , B 0 A, B C A, B A, C Copyright – Michael D. Fayer, 2007 Expectation Value and Averages A a a eigenvector normalized eigenvalue If make measurement of observable A on state a will observe . What if measure observable A on state not an eigenvector of operator A. Ab ? normalized Expand b in complete set of eigenkets a Superposition principle. Copyright – Michael D. Fayer, 2007 b c1 a1 c2 a2 c3 a3 (If continuous range integral) b ci a i i Consider only two states (normalized and orthogonal). b c1 a1 c2 a2 A b A c1 a1 c2 a2 c1 A a1 c2 A a2 1c1 a1 2c2 a2 Left multiply by b . b A b c1* a1 c2* a2 c 1 1 a1 2c2 a2 1c1*c1 2c2*c2 1 c1 2 c2 2 2 Copyright – Michael D. Fayer, 2007 The absolute square of the coefficient ci, | ci|2, in the expansion of b in terms of the eigenvectors ai of the operator (observable) A is the probability that a measurement of A on the state b will yield the eigenvalue i. If there are more than two states in the expansion b ci a i i b A b i ci 2 i eigenvalue probability of eigenvalue Copyright – Michael D. Fayer, 2007 Definition: The average is the value of a particular outcome times its probability, summed over all possible outcomes. Then b A b ci i 2 i is the average value of the observable when many measurements are made. Assume: One measurement on a large number of identically prepared non- interacting systems is the same as the average of many repeated measurements on one such system prepared each time in an identical manner. Copyright – Michael D. Fayer, 2007 b Ab Expectation value of the operator A. In terms of particular wavefunctions b Ab b A bd Copyright – Michael D. Fayer, 2007 The Uncertainty Principle - derivation Have shown - x , P 0 and that x p Want to prove: Given A and B, Hermitian with A , B i C C another Hermitian operator (could be number – special case of operator, identity operator). Then 1 AB C 2 with C SC S short hand for expectation value S and S arbitrary but normalized. Copyright – Michael D. Fayer, 2007 Consider operator D A B i B arbitrary real numbers D S Q Q Q S DD S 0 Since Q Q is the scalar product of vector with itself. Q Q S DD S A2 2 2 B 2 C ' C 0 (derive this in home work) C ' AB BA is the anticommutator of A and B . AB BA A , B 2 A S A S anticommutator 2 Copyright – Michael D. Fayer, 2007 B S 0 for arbitrary ket S . Can rearrange to give 1 C' 2 2 A B 2 B2 2 2 2 B 1 C 2 B2 1 C' 4 B2 2 2 1 C 0 2 4 B Holds for any value of and . Pick and so terms in parentheses are zero. Then A2 B 2 1 C 4 2 C' 2 1 C 4 2 (Have multiplied through by 2 B and transposed.) Positive numbers because square of real numbers. The sum of two positive numbers is one of them. Thus, A 2 B 2 1 C 4 2 Copyright – Michael D. Fayer, 2007 A 2 B Define 2 1 C 4 A B 2 2 A A 2 B 2 2 2 B Second moment of distribution - for Gaussian standard deviation squared. 2 For special case A B 0 1 AB C 2 Average value of the observable is zero. C 1 2 2 C square root of the square of a number Copyright – Michael D. Fayer, 2007 Have proven that for A , B i C 1 AB C 2 Example x, P i Number, special case of an operator. Number is implicitly multiplied by the identity operator. x P 0 Therefore x p / 2. Uncertainty comes from superposition principle. The more general case is discussed in the book. Copyright – Michael D. Fayer, 2007