Presentation on theme: "The Measuring Process:"— Presentation transcript:
1 The Measuring Process: a Superpositionora Mixture?
2 The 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.)
3 The 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 :
4 The 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
5 The Stern-Gerlach Spectroscopy The PhenomenonA beam of oxygen molecules passing through a non-homogeneous magnetic field splits into a number of beamsz
6 The 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
7 The 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)
8 Filtering 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)
9 Measuring 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.
10 Separating 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
11 The 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
12 Superposition 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
13 How 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
14 An 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
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