The Measuring Process: a Superposition or a Mixture?

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.)

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  :

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 

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

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

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)

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)

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.

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

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

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

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

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