Quantum Physics Mathematics. Quantum Physics Tools in Real Life Reality.

Quantum PhysicsMathematics

Quantum PhysicsMathematics

Quantum PhysicsTools in Real LifeQuantum PhysicsTools in Real Life

Reality

Quantum PhysicsTools in Physics / Quantum PhysicsReal Number – Vector - Statevector

Quantum PhysicsTools in Physics / Quantum PhysicsReal Number – Vector - Statevector

Speed represented by a real number80 km/h

Velocity represented by a vector 80 km/h NorthEast

State representered by a statevector

v

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Quantum PhysicsTools in Physics / Quantum PhysicsQuantum PhysicsTools in Physics / Quantum Physics

Reality

Language

Position is 2.0 m andvelocity is 4.0 m/s

Mathematics

Numbers Variables

Vectors

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Quantum PhysicsSuperpositionVectors - Functions

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Heat Sampling

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Quantum PhysicsReality - Theory / Mathematical roomQuantum PhysicsReality - Theory / Mathematical room

Reality

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Quantum PhysicsReality - Theory / Mathematical room - Classical PhysicsQuantum PhysicsReality - Theory / Mathematical room - Classical Physics

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Quantum PhysicsPostulate 1Quantum PhysicsPostulate 1

)(t

1.Every system is described by a state vectorthat is an element of a Hilbert space.

Quantum PhysicsPostulate 2Quantum PhysicsPostulate 2

2.An action or a measurement on a systemis associated with an operator.

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2

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Quantum PhysicsObservation / Measurement in daily lifeQuantum PhysicsObservation / Measurement in daily life

The behaviour of the classis perhaps not independentof an observation (making a video of the class)

The length of the tableis independent of an observation or a measurement.

2

1

Quantum PhysicsObservation of position, changing the velocityQuantum 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.

2

1

Quantum PhysicsObservation of current and voltageQuantum PhysicsObservation of current and voltage

2

1

Input of amperemeter and voltmeterdisturb the current and voltage.

Quantum PhysicsStern-Gerlach experiment [1/2]Observation of angular momentum in one directioninfluence on the angular momentum in another direction


Silver atoms going through a vertical magnetic fielddividing the beam into two new beamsdependent of the angular momentum of the atom.

Three magnetic fields: Vertical, horizontal, vertical.Every time the beam is divided into two new beams.

No sorting mechanism.A new vertical/horisontal measurementdisturbs/changes the horisontal/vertical beam property.

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Quantum PhysicsEntanglementQuantum PhysicsEntanglement

Quantum entanglement occurs when particles such asphotons, electrons, molecules and even small diamondsinteract 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 foundto have taken the appropriate correlated value.

2

1

Entanglement is a challange in our understanding of natureog will hopefully give us new technological applications.

According to the Copenhagen interpretation of quantum physics,their shared state is indefinite until measured.

Quantum PhysicsObservation / Measurement - ClassicalQuantum 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.

Quantum PhysicsObservation / Measurement - QuantumQuantum PhysicsObservation / Measurement - Quantum

A car is placed in the position Abehind a person.The person with the car behind,cannot yet observe the car.

The person turns aroundand observeres the carin the position B.

In quantum physicsit’s possible that the observation of a property of the carmoves the car to another position.

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B

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Quantum PhysicsObservation / Measurement - QuantumQuantum PhysicsObservation / Measurement - Quantum

Question:Where was the carbefore the observation?

Realist:The car was at B.

If this is true,quantum physics is incomplete.There must be some hidden variables (Einstein).

?

B

Orthodox:The car wasn’t really anywhere.

Agnostic:Refuse to answer.

It’s the act of measurementthat force the particle to ‘take a stand’.Observations not only disturb,but they also produce.

No sense to ask before a measurent.

Orthodox supported by theory (Bell 1964) and experiment (Aspect 1982).

Quantum PhysicsObservationQuantum PhysicsObservation

Before the measurement

After the measurement

1

2

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The measurement producea specific position of the car.

A repeated measurementon the new systemproduce the same result.

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Quantum PhysicsObservationSuperposition

Quantum PhysicsObservationSuperposition

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Quantum PhysicsSuperpositionFourier

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Quantum PhysicsClassical:Vector expanded in an orthonormal basis - I

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Quantum PhysicsClassical:Vector expanded in an orthonormal basis - II

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Quantum PhysicsSpace - Dual spaceIntroduction

Quantum PhysicsSpace - Dual spaceIntroduction

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Quantum PhysicsSpace - Dual spaceExample - Real Elements

Quantum PhysicsSpace - Dual spaceExample - Real Elements

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Quantum PhysicsSpace - Dual spaceExample - Complex Elements

Quantum PhysicsSpace - Dual spaceExample - Complex Elements

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Quantum PhysicsSpace - Dual spaceExample - Scalar Product

Quantum PhysicsSpace - Dual spaceExample - Scalar Product

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Quantum PhysicsSpace - Dual spaceExample - Operator I

Quantum PhysicsSpace - Dual spaceExample - Operator I

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Quantum PhysicsSpace - Dual spaceExample - Operator II

Quantum PhysicsSpace - Dual spaceExample - Operator II

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Quantum PhysicsSpace - Dual spaceExample - Operator III

Quantum PhysicsSpace - Dual spaceExample - Operator III

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Quantum PhysicsProbability amplitudeQuantum PhysicsProbability amplitude

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Quantum PhysicsProjection OperatorTheory

Quantum PhysicsProjection OperatorTheory

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Quantum PhysicsProjection OperatorExample - Projection to basisfunction

Quantum PhysicsProjection OperatorExample - Projection to basisfunction

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Quantum PhysicsProjection OperatorExample - Projection to subroom

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Quantum PhysicsUnit OperatorTheory

Quantum PhysicsUnit OperatorTheory

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Quantum PhysicsUnit OperatorExample

Quantum PhysicsUnit OperatorExample

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Quantum PhysicsOrthonormality - CompletenessDiscrete - Continuous

Quantum PhysicsOrthonormality - CompletenessDiscrete - Continuous

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Quantum PhysicsOperatorEigenvectors - Eigenvalues

Quantum PhysicsOperatorEigenvectors - Eigenvalues

Eigenvectors are of special interest since experimentally we always observethat subsequent measurements of a system return the same result (collapse of wave function).

Consequence of Spectral Theorem:

The only allowed physical results of measurements of the observable Aare the elements of the spectrum of the operator which corresponds to A.

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Quantum PhysicsOperatorSelf-adjoint operator

Quantum PhysicsOperatorSelf-adjoint operator

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Quantum PhysicsOperatorTheorem

Quantum PhysicsOperatorTheorem

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Quantum PhysicsHermitian operatorThe eigenvalues of a Hermitian operator are real

Quantum PhysicsHermitian operatorThe eigenvalues of a Hermitian operator are real

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Quantum PhysicsHermitian operatorEigenstates with different eigenvalues are orthogonal

Quantum PhysicsHermitian operatorEigenstates with different eigenvalues are orthogonal

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Quantum PhysicsOperatorexpanded by eigenvectors

Quantum PhysicsOperatorexpanded by eigenvectors

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Quantum PhysicsAverage of Operator [1/2]Quantum PhysicsAverage of Operator [1/2]

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Quantum PhysicsDetermiate stateQuantum PhysicsDetermiate state

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The determinate state of the operator Athat return the same value a every timeis the eigenstate of A with the eigenvalue a.

Unless the state is an eigenstate of the actual operator,we can never predict the result of the operatoronly the probability.

Quantum PhysicsUncertainty [1/3]Quantum PhysicsUncertainty [1/3]

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Quantum PhysicsKinematics and dynamicsQuantum PhysicsKinematics and dynamics

Necessary with correspondence rulesthat identify variables with operators.

This can be done by studying specialsymmetries and transformations. aA

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The laws of nature are believed to be invariantunder certain space-time symmetry operations,including displacements, rotations,and transformations between frames of reference.

Corresponding to each such space-time transformationthere must be a transformation of observables,opetators and states.

Energy, linear and angular momentumare closely related to space-time symmetry transformations.

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Quantum PhysicsNoether’s theoremQuantum PhysicsNoether’s theorem

Symmetry Conservation

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Quantum PhysicsInvariant transformationUnitary operator

Quantum PhysicsInvariant transformationUnitary operator

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Quantum PhysicsUnitary operatorQuantum PhysicsUnitary operator

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Quantum PhysicsGenerator of infinitesimal transformationQuantum PhysicsGenerator of infinitesimal transformation

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Quantum PhysicsTime operator - Time dependent Schrödinger equationEnergy operator - Generator for time displacement

Quantum PhysicsTime operator - Time dependent Schrödinger equationEnergy operator - Generator for time displacement

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Quantum PhysicsUncertaintyPosition / Momentum - Energy / Time

Quantum PhysicsUncertaintyPosition / Momentum - Energy / Time

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Quantum Physics Mathematics. Quantum Physics Tools in Real Life Reality.

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Transcript of Quantum Physics Mathematics. Quantum Physics Tools in Real Life Reality.