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Quantum Physics Mathematics

Quantum Physics Mathematics. Quantum Physics Tools in Real Life. Reality. Quantum Physics Tools in Physics / Quantum Physics Real Number – Vector - Statevector. Speed represented by a real number. 80 km/h. Velocity represented by a vector. 80 km/h Nor t hEast.

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Quantum Physics Mathematics

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  1. Quantum PhysicsMathematics

  2. Quantum PhysicsTools in Real Life Reality

  3. Quantum PhysicsTools in Physics / Quantum PhysicsReal Number – Vector - Statevector Speed represented by a real number 80 km/h Velocity represented by a vector 80 km/h NorthEast State representered by a statevector

  4. Quantum PhysicsTools in Physics / Quantum Physics Mathematics Language Numbers Variables Functions Position is 2.0 m and velocity is 4.0 m/s Vectors Reality

  5. Quantum PhysicsSuperpositionVectors - Functions S 1 3 2 4 Music Pulse train Heat Sampling

  6. Quantum PhysicsReality - Theory / Mathematical room Theory / Mathematical room Reality

  7. Quantum PhysicsReality - Theory / Mathematical room - Classical Physics Theory Mathematical room Reality

  8. Quantum PhysicsReality - Theory / Mathematical room - Quantum Physics Theory Mathematical room Reality

  9. Quantum PhysicsPostulate 1 1. Every system is described by a state vector that is an element of a Hilbert space.

  10. Quantum PhysicsPostulate 2 2. An action or a measurement on a system is associated with an operator.

  11. Quantum PhysicsObservation / Measurement in daily life The length of the table is independent of an observation or a measurement. The behaviour of the class is perhaps not independent of an observation (making a video of the class)

  12. Quantum PhysicsObservation of position, changing the velocity A ball with a known velocity and unknown position. Try to determine the position. A bit unlucky one foot hits the ball. The position is known when the ball is touched, but now the velocity is changing. Just after the hit of the ball, the position is known, but now the velocity is unknown.

  13. Quantum PhysicsObservation of current and voltage Input of amperemeter and voltmeter disturb the current and voltage.

  14. Quantum PhysicsStern-Gerlach experiment [1/2]Observation of angular momentum in one directioninfluence on the angular momentum in another direction V1 Silver atoms going through a vertical magnetic field dividing the beam into two new beams dependent of the angular momentum of the atom. B1 B B2 V2 V1 Three magnetic fields: Vertical, horizontal, vertical. Every time the beam is divided into two new beams. No sorting mechanism. A new vertical/horisontal measurement disturbs/changes the horisontal/vertical beam property. H1 B11 B121 B1 B12 B B122 B2

  15. Quantum PhysicsStern-Gerlach experiment [2/2]Observation of angular momentum in one directioninfluence on the angular momentum in another direction z+ z+ S-G z-axis S-G z-axis 1 B z- No z- x+ z+ S-G z-axis S-G x-axis 2 B x- z- z+ x+ z+ S-G z-axis S-G x-axis S-G z-axis 3 B z- x- z-

  16. Quantum PhysicsEntanglement Quantum entanglement occurs when particles such as photons, electrons, molecules and even small diamonds interact physically and then become separated. When a measurement is made on one of member of such a pair, the other member will at any subsequent time be found to have taken the appropriate correlated value. According to the Copenhagen interpretation of quantum physics, their shared state is indefinite until measured. Entanglement is a challange in our understanding of nature og will hopefully give us new technological applications.

  17. Quantum PhysicsObservation / Measurement - Classical A car (particle) is placed behind a person. The person with the car behind, cannot see the car. The person turns around and observeres the car. Classically we will say: The car was at the same place also just before the observation.

  18. Quantum PhysicsObservation / Measurement - Quantum A A car is placed in the position A behind a person. The person with the car behind, cannot yet observe the car. The person turns around and observeres the car in the position B. B In quantum physics it’s possible that the observation of a property of the car moves the car to another position.

  19. Quantum PhysicsObservation / Measurement - Quantum Question: Where was the car before the observation? ? Realist: The car was at B. If this is true, quantum physics is incomplete. There must be some hidden variables (Einstein). Orthodox: The car wasn’t really anywhere. It’s the act of measurement that force the particle to ‘take a stand’. Observations not only disturb, but they also produce. B Agnostic: Refuse to answer. No sense to ask before a measurent. Orthodox supported by theory (Bell 1964) and experiment (Aspect 1982).

  20. Quantum PhysicsObservation M Before the measurement M After the measurement

  21. M M Quantum PhysicsObservationSuperposition Before the measurement the position of the car is a superposition of infinitely many positions. M The measurement produce a specific position of the car. M A repeated measurement on the new system produce the same result.

  22. Quantum PhysicsSuperpositionFourier Music Pulse train Heat Sampling

  23. Quantum PhysicsClassical:Vector expanded in an orthonormal basis - I

  24. Quantum PhysicsClassical:Vector expanded in an orthonormal basis - II Complex coefficients

  25. Quantum PhysicsState vector expanded in an orthonormal basis

  26. Quantum PhysicsSpace - Dual spaceIntroduction Ket Bra Dual space Space

  27. Quantum PhysicsSpace - Dual spaceExample - Real Elements Ket Bra Dual space Space

  28. Quantum PhysicsSpace - Dual spaceExample - Complex Elements Bra Ket Ket Bra Dual space Space

  29. Quantum PhysicsSpace - Dual spaceExample - Scalar Product Ket Bra Dual space Space

  30. Quantum PhysicsSpace - Dual spaceExample - Operator I Ket Bra Dual space Space

  31. Quantum PhysicsSpace - Dual spaceExample - Operator II Ket Bra Dual space Space

  32. Quantum PhysicsSpace - Dual spaceExample - Operator III Ket Bra Dual space Space

  33. Quantum PhysicsProbability amplitude Same state Normalization of a state vector don’t change the probability distributions. Therefore we postulate c and  to represent the same state. cn : Probability amplitude cn2 : Probability

  34. Quantum PhysicsProjection OperatorTheory

  35. Quantum PhysicsProjection OperatorExample - Projection to basisfunction

  36. Quantum PhysicsProjection OperatorExample - Projection to subroom

  37. Quantum PhysicsUnit OperatorTheory

  38. Quantum PhysicsUnit OperatorExample

  39. Quantum PhysicsOrthonormality - Completeness Orthonormality Projection Operator Completeness

  40. Quantum PhysicsOrthonormality - CompletenessDiscrete - Continuous Continuous Discrete State Orthonormality Projection Operator Completeness

  41. Quantum PhysicsOperatorEigenvectors - Eigenvalues Eigenvectors are of special interestsince experimentally we always observe that subsequent measurements of a systemreturn the same result (collapse of wave function). A A Consequence of Spectral Theorem: The only allowed physical results of measurements of the observable A are the elements of the spectrum of the operatorwhich corresponds to A. Measured quantity

  42. Quantum PhysicsOperatorSelf-adjoint operator Def: Self-adjoint operator: Def: Hermitian operator: The distinction between Hermitian and self-adjoint operators is relevant only for operators in infinite-dimensional vector spaces. Proof:

  43. Quantum PhysicsOperatorTheorem Theorem: Proof: Canceling i and adding

  44. Quantum PhysicsHermitian operatorThe eigenvalues of a Hermitian operator are real Theorem: The eigenvalues of a Hermitian operator are real. Proof:

  45. Quantum PhysicsHermitian operatorEigenstates with different eigenvalues are orthogonal Theorem: Eigenstates corresponding to distinct eigenvalues of an Hermitian operator must be orthogonal. Proof:

  46. Quantum PhysicsOperatorexpanded by eigenvectors Eigenvectors are of special interest since experimentally we always observe that subsequent measurements of a system return the same result (collapse of wave function) The measurable quantity is associated with the eigenvalue. This eigenvalue should be real so A have to be a self-adjoint operator A+ = A Every operator can be expanded by their eigenvectors and eigenvalues

  47. Quantum PhysicsAverage of Operator [1/2]

  48. Quantum PhysicsAverage of Operator [2/2]

  49. Quantum PhysicsDetermiate state Determinate state: A state prepared so a measurement of operator A is certain to return the same value a every time. Unless the state is an eigenstate of the actual operator, we can never predict the result of the operator only the probability. The determinate state of the operator A that return the same value a every time is the eigenstate of A with the eigenvalue a.

  50. Quantum PhysicsUncertainty [1/3]

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