Quantum Physics Mathematics

Quantum Physics Tools in Real Life Reality

Quantum Physics Tools in Physics / Quantum Physics Reality Language Position is 2.0 m and velocity is 4.0 m/s Mathematics NumbersVariables Vectors Functions

Quantum Physics Superposition Vectors - Functions Music Pulse train Heat Sampling S

Quantum Physics Reality - Theory / Mathematical room Reality Theory / Mathematical room

Quantum Physics Reality - Theory / Mathematical room - Classical Physics Reality Theory Mathematical room

Quantum Physics Reality - Theory / Mathematical room - Quantum Physics Reality Theory Mathematical room

Quantum Physics Theories DoDo DnDn D o = Domain described by old theory D n = Domain described by new theory DoDo DnDn Ordinary Classic Theory Quantum Physics Relativity Theory Micro Macro High velocity Low velocity

Quantum Physics Postulates 1. Every system is described by a state vector that is an element of a Hilbert space. 2. An action or a measurement on a system is associated with an operator.

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

Quantum Physics Observation 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.

Quantum Physics Observation of current and voltage Input of amperemeter and voltmeter disturb the current and voltage.

Quantum Physics Observation of angular momentum in one direction influence on the angular momentum in another direction Silver atoms going through a vertical magnetic field dividing the beam into two new beams dependent 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 measurement disturbs/changes the horisontal/vertical beam property.

Quantum Physics Entanglement 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. Entanglement is a challange in our understanding of nature og will hopefully give us new technological applications. According to the Copenhagen interpretation of quantum physics, their shared state is indefinite until measured.

Quantum Physics Observation / 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 Physics Observation / Measurement - Quantum 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. In quantum physics it’s possible that the observation of a property of the car moves the car to another position. A B

Quantum Physics Observation / 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). ? B Orthodox: The car wasn’t really anywhere. Agnostic: Refuse to answer. It’s the act of measurement that 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 Physics Observation Before the measurement After the measurement M M

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

Quantum Physics Superposition Fourier Music Pulse train Heat Sampling

Quantum Physics Classical: Vector expanded in an orthonormal basis - I

Quantum Physics Classical: Vector expanded in an orthonormal basis - II Complex coefficients

Quantum Physics State vector expanded in an orthonormal basis

Quantum Physics Space - Dual space SpaceDual space Bra Ket

Quantum Physics Probability amplitude c n : Probability amplitude c n 2 : Probability Normalization of a state vector don’t change the probability distributions. Therefore we postulate c  and  to represent the same state. Same state

Quantum Physics Projection Operator

Quantum Physics Unit Operator

Quantum Physics Orthonormality - Completeness Orthonormality Projection Operator Completeness

Quantum Physics Orthonormality - Completeness Discrete - Continuous Orthonormality Projection Operator Completeness State Discrete Continuous

Quantum Physics Operator Eigenvectors - Eigenvalues Eigenvectors are of special interest since experimentally we always observe that 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 A are the elements of the spectrum of the operator which corresponds to A. AA Measured quantity

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

Quantum Physics Operator Theorem Theorem: Proof: Canceling i and adding

Quantum Physics Hermitian operator The eigenvalues of a Hermitian operator are real Theorem: Proof: The eigenvalues of a Hermitian operator are real.

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

Quantum Physics Operator expanded by eigenvectors Every operator can be expanded by their eigenvectors and eigenvalues 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

Quantum Physics Average of Operator [1/2]

Quantum Physics Average of Operator [2/2]

Quantum Physics Determiate state Determinate state: A state prepared so a measurement of operator A is certain to return the same value a every time. The determinate state of the operator A that return the same value a every time is 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 operator only the probability.

Quantum Physics Uncertainty [1/3]

Quantum Physics Uncertainty [2/3]

Quantum Physics Uncertainty [3/3]

Quantum Physics Kinematics and dynamics Necessary with correspondence rules that identify variables with operators. This can be done by studying special symmetries and transformations. Operator  Variable The laws of nature are believed to be invariant under certain space-time symmetry operations, including displacements, rotations, and transformations between frames of reference. Corresponding to each such space-time transformation there must be a transformation of observables, opetators and states. Energy, linear and angular momentum are closely related to space-time symmetry transformations.

Quantum Physics Noether’s theorem SymmetryConservation Time translation EnergyE Space translation Linear momentump RotationAngular momentumL Noether’s (first) theorem states that any differentiable symmetry of the action (law) of a physical system has a corresponding conservation law.

Quantum Physics Invariant transformation Unitary operator Any mapping of the vector space onto itself that preserves the value of | | may be implemented by an operator U being either unitary (linear) or antiunitary (antilinear). Only linear operators can describe continous transformations. The laws of nature are believed to be invariant under certain space-time symmetry. Therefore we are looking for a continous transformation that are preservering the probability distribution.

Quantum Physics Unitary operator Any mapping of the vector space onto itself that preserves the value of | | may be implemented by an operator U being ether unitary (linear) or antiunitary (antilinear). Only linear operators can describe continous transformations. Only linear operators can describe continous transformations.

Quantum Physics Generator of infinitesimal transformation Any mapping of the vector space onto itself that preserves the value of | | may be implemented by an operator U being ether unitary (linear) or antiunitary (antilinear). Only linear operators can describe continous transformations.

Quantum Physics Time operator - Time dependent Schrödinger equation Energy operator - Generator for time displacement Energy operator Generator for time displacement

Quantum Physics Schrödinger equation - Time independent Stationary states

Quantum Physics Normalization is time-independent

Quantum Physics Moment operator in position space

Quantum Physics Schrödinger equation - Time independent

Quantum Physics Operators Position Potensial energy Momentum Kinetic energy Total energy Position spaceMomentum space

Quantum Physics Uncertainty Position / Momentum - Energy / Time Position Momentum Energy Time