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**The Measuring Process:**

a Superposition or a Mixture?

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**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) Or The system is composed of a single particle with reduced mass, moving around the center of gravity (precise.)

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**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 :

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**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 (- )F y x A particle with a definite angular momentum. There is a complete uncertainty of the angle A particle with a preferred axis in space. There is a complete uncertainty of the angular momentum x y

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**The Stern-Gerlach Spectroscopy**

The Phenomenon A beam of oxygen molecules passing through a non-homogeneous magnetic field splits into a number of beams z

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**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. S N

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**The Stern-Gerlach Spectroscopy**

A Quantum Model The angular momentum is a singular measurement. Each particle is moving along one of the orbits according to its angular momentum. N S A mixture (each particle in a different quantum state)

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**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: N S m=1 50% m=-1 50% Superposition (one particle in two states)

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**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.

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**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 momentum A basis state for measuring polarization (direction in space) y x

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**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

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**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

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**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 directions Superposition: a preference to an aignement in the x axis direction

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**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-H Li

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Ch 9 pages 469-476 Lecture 23 – The Hydrogen Atom.

Ch 9 pages 469-476 Lecture 23 – The Hydrogen Atom.

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