Presentation on theme: "The Measuring Process:"— Presentation transcript:
1The Measuring Process: a Superpositionora Mixture?
2The Rotational Motion of a Diatomic Molecule Upon first approximation let us regard a molecular rotating system as a free particle on a ring (a two-dimensional hard rotor.)The working assumptions are:The molecule is moving on a plane (approximation)The molecular radius is rigid (approximation)The central atom is stationary (approximation)OrThe system is composed of a single particle with reduced mass, moving around the center of gravity (precise.)
3The Postulates of Quantum Mechanics for a Free Particle on a Ring (The game tools) The state of the system can be described by a wavepacket, (the game board) belonging to the space of continuous functions in angle :(The game rules) For each component in the wavepacket, the following is true:(The interface) the measuring operation has the following probability of finding the particle in the angle element d :
4The Representation of a Free Particle on a Ring The system state is defined for a classical particle by the particle’s angle and its angular momentum.In quantum mechanics the system state is defined by a wave function that has a complex value for each angle. For example:(+ )H(- )FyxA particle with a definite angular momentum. There is a complete uncertainty of the angleA particle with a preferred axis in space. There is a complete uncertainty of the angular momentumxy
5The Stern-Gerlach Spectroscopy The PhenomenonA beam of oxygen molecules passing through a non-homogeneous magnetic field splits into a number of beamsz
6The Stern-Gerlach Spectroscopy A Classical Model (for a planetary HF)The motion of the positively charged hydrogen is equivalent to a ring current. The current inspires a magnetic moment µB relative to the angular momentum, the external field Bz exerts force Bz on the magnetic moment.SN
7The Stern-Gerlach Spectroscopy A Quantum ModelThe angular momentum is a singular measurement. Each particle is moving along one of the orbits according to its angular momentum.NSA mixture (each particle in a different quantum state)
8Filtering a Superposition The measuring system is a filter that separates basis states. While measuring, each particle appears in one point only. The probability for this is:NSm=1 50%m=-1 50%Superposition (one particle in two states)
9Measuring as a Destructive Process The Uncertainty Principle Measuring results in a certainty of one property in exchange to an uncertainty of another one, which had been known prior to the measuring process.Before measurement: The particle is in the x axis direction, the angular momentum is unknown.After measurement: The angular momentum is known, the particle’s direction is unknown.
10Separating to Basis States Basis State: a quantum state with a well-defined particle property (position, momentum, angle, angular momentum, polarization, energy, etc.) A set of basis states is measured for each dimension. A basis state for one measurement is not necessarily a basis for another.A basis state for measuring the angular momentumA basis state for measuring polarization (direction in space)yx
11The Completeness of the Basis (Fourier Theorem) Each periodic function defined on [ , -] can be decomposed linearly:-In this way it is possible to calculate the probability of finding the particle in a given angular momentum for each continuous function on the ring
12Superposition Versus Mixture The wave function in a superposition is simultaneously in various quantum states. The measuring process causes each particle to choose only one basis state and this results in a mixture.After measuring Before measuring
13How Can We Distinguish Between the Two? By measuring another property: the probability of a reaction in different angles. The angular dependency of the steric factor in a nucleofilic charge reaction is examined. The basis set of the direction measurements differentiates between superposition and a mixture of states of angular momentum.Mixture: a homogeneous distribution in all directionsSuperposition: a preference to an aignement in the x axis direction
14An Experiment with Crossing Beams The reaction Li + HF LiF + H is examined by crossing beams of reactants and measuring the amount of the output in different angles. The HF molecule is aligned in the Px state.H-F /F-HLi