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Transcript

The Measuring Process: a Superposition or a Mixture? 1 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 twodimensional hard rotor.) The working assumptions are: 1. The molecule is moving on a plane (approximation) 2. The molecular radius is rigid (approximation) 3. The central atom is stationary (approximation) Or 4. The system is composed of a single particle with reduced mass, moving around the center of gravity (precise.) 2 The Postulates of Quantum Mechanics for a Free Particle on a Ring 1. (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 : ( , t ) g (m) m ( , t ) ; m ( ,0) 1 2 m e im 2. (The game rules) For each component in the wavepacket, the following is true: m ( , t ) e E i t m ( ,0) ; E (m) / 2 I 2 3. (The interface) the measuring operation has the following probability of finding the particle in the angle element d : ( , t ) d 2 3 The Representation of a Free Particle on a Ring y 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: y y A particle with a preferred axis in space. There is a complete uncertainty of the x angular momentum ( ) x 1 2 px [ 1 ( ) 1 ( )] y (+ )H x (- )F A particle with a definite angular momentum. x There is a complete uncertainty of the angle y 2 ( ) x 1 2 ei 2 m2 Lz 2 4 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 5 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. B Lz B Lz Bz N Fz Bz B S 6 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. Lz m N 2 0 A mixture (each particle in a different quantum state) S 2 7 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: P(m) g ( m) 2 1 ( ) px 1 2 N [ 1 ( ) 1 ( )] Superposition (one particle in two states) S 1 ( ) m=1 50% m=-1 50% 8 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. Lz h 9 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 polarization (direction in space) A basis state for measuring the angular momentum y y x x 10 The Completeness of the Basis (Fourier Theorem) Each periodic function defined on [ , -] can be decomposed linearly: ( ) g (m) m ( ) m m ( ) 1 2 e im ; g (m) m ( ) ( ) d * 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 - 0 11 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 1 2 2 1 2 2 Before measuring 1 1 2 2 2 12 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 1 2 1 2 2 2 2 2 13 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 H-F /F-H Li 14