MIT OpenCourseWarehttp://ocw.mit.edu 6.453 Quantum Optical Communication Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.September 11, 2008Optical and Quantum Communications Groupwww.rle.mit.edu/qoptics6.453 Quantum Optical CommunicationLecture 3Jeffrey H. Shapiro2www.rle.mit.edu/qoptics6.453 Quantum Optical Communication — Lecture 3! Announcements! Turn in problem set 1! Pick up problem set 1 solution, problem set 2, lecture notes, slides! Fundamentals of Dirac-Notation Quantum Mechanics! Definitions and axioms — reprise! Quantum measurements — statistics! Schrödinger picture versus Heisenberg picture! Heisenberg uncertainty principle3www.rle.mit.edu/qopticsQuantum Systems and Quantum States! Definition 1:A quantum-mechanical system is a physical systemgoverned by the laws of quantum mechanics.! Definition 2:The state of a quantum mechanical system at a particulartime is the sum total of all information that can be knownabout the system at time . It is a ket vector in anappropriate Hilbert space of possible states. Finiteenergy states have unit length ket vectors, i.e., .4www.rle.mit.edu/qopticsTime Evolution via the Schrödinger Equation! Axiom 1:For , an isolated system with initial state will reachstatewhere is the unitary time-evolution operator for thesystem . is obtained by solvingwhere is the Hamiltonian (energy) operator for .Equivalently, we have the Schrödinger equation5www.rle.mit.edu/qopticsQuantum Measurements: Observables! Axiom 2:An observable is a measurable dynamical variable of thequantum system . It is represented by an Hermitianoperator which has a complete set of eigenkets.! Axiom 3:For a quantum system that is in state at time ,measurement of the observableyields an outcome that is one of the eigenvalues, , with6www.rle.mit.edu/qopticsQuantum Measurements: Observables! Projection postulate:Immediately after a measurement of an observable , withdistinct eigenvalues, yields outcome the state of thesystem becomes .! Axiom 3a:For a quantum system that is in state at time ,measurement of the observableyields an outcome that is one of the eigenvalues, , with7www.rle.mit.edu/qopticsQuantum Measurements: Statistics! Average Value of an Observable Measurement! Discrete eigenvalue spectrum! Continuous eigenvalue spectrum! Variance of an Observable Measurement8www.rle.mit.edu/qopticsSchrödinger versus Heisenberg Pictures! Schrödinger Picture! Observables are time-independent operators! Between measurements, states evolve according to the Schrödingerequation! Heisenberg Picture! Between measurements, states are constant! Observables evolve in time according to appropriate equations ofmotion9www.rle.mit.edu/qopticsConverting Between Pictures! Statistics of an Observable Measurement! Schrödinger picture! Heisenberg picture! Invariance of Statistics to Choice of Picture10www.rle.mit.edu/qopticsHeisenberg Equations of Motion! Transforming an Observable between Pictures! Equation of Motion for! Commutator Brackets11www.rle.mit.edu/qopticsHeisenberg Uncertainty Principle! and Noncommuting Observables! Lower Limit on Product of Individual Measurement Variances12www.rle.mit.edu/qopticsComing Attractions: Lectures 4 and 5! Lecture 4:Quantum Harmonic Oscillator! Quantization of a classical LC circuit! Annihilation and creation operators! Energy eigenstates — number-state kets! Lecture 5:Quantum Harmonic Oscillator! Quadrature measurements! Coherent