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1 The Quantization of the Angular Momentum

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2 In the gas phase discrete absorption lines appear in the spectral reagions where in the liquid phase the absorbtion is continuously. Quantization of the absorption In the gas phase, unlike the liquid phase there are additional free translation and rotation degrees of freedom. The rotation* is the one responsible for the absorption lines. * The rotation is the degree of freedom of a free particle, and therefore has a continuous energy (k is continuous) The Phenomenon The Model CH 3 I ( ) CH 3 I (g)

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3 (- ) I (+ ) CH 3 zx y A free Particle on a Ring – a Classical picture A system composed of two particles, which are connected by a rigid rod of length r. The particles perform a rotating motion on a plain. This system is equivalent to a single particle of a reduced mass µ, moving around the center of mass, with a constant radius.

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4 The Quantum Mechanics Postulates for a Free Particle on a Ring 1. (The tools of the game) The system state can be described by a wavepacket, (the board of the game) pertaining to the space of continuous functions in angle : 2. (The rules of the game) For each component in the wavepacket the following is true: 3. (The interface) the measurement outcome has the following probability of finding the particle in an angle :

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5 m=1m=1.5m=2 Postulates I: Quantization of m To meet the continuity condition it is possible to include in the wavepacket only functions whose quantum number m is an integer

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6 Postulates II: Quantization of the Angular Momentum and the Energy The dispersion ratio is selected to ccorrospond to the classical limit of h: To each wave are attributed the angular momentum and the kinetic energy

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7 Postulates III : Dipole Moment In a single wave the charge distribution is symmetrical. In superposition of two waves it is possible to obtain an asymmetrical distribution of the charge, which is equivalent to an existence of a Dipole Moment += y y y x xx

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8 Quantization of the Light Absorption A rotating Dipole Moment is capable of exchanging energy with a radiation field in its self frequency (the resonance principle.) The frequency of an envelope changing in a superposition is: And therefore:

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