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The Quantization The Quantization of the of the

Angular MomentumAngular Momentum

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In the gas phase discrete In the gas phase discrete absorption lines appear in the absorption lines appear in the spectral reagions where in the spectral reagions where in the liquid phase the absorbtion is liquid phase the absorbtion is continuously.continuously.

Quantization of the absorptionQuantization of the absorption

In the gas phase, unlike the liquid In the gas phase, unlike the liquid phase there are additional free phase there are additional free translation and rotation degrees of translation and rotation degrees of freedom. The rotation* is the one freedom. The rotation* is the one responsible for the absorption responsible for the absorption lines. lines.

* The rotation is the degree of freedom of a free particle, and therefore * The rotation is the degree of freedom of a free particle, and therefore has a continuous energy (k is continuous)has a continuous energy (k is continuous)

* The rotation is the degree of freedom of a free particle, and therefore * The rotation is the degree of freedom of a free particle, and therefore has a continuous energy (k is continuous)has a continuous energy (k is continuous)

The PhenomenonThe Phenomenon The ModelThe Model

CHCH33II(())CHCH33II(g)(g)

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-(-())II+(+())CHCH33

zz

xx

yy

A free Particle on a Ring – a Classical pictureA free Particle on a Ring – a Classical picture

A system composed of two particles, which are connected A system composed of two particles, which are connected by a rigid rod of length r. The particles perform a rotating by a rigid rod of length r. The particles perform a rotating motion on a plain. This system is equivalent to a single motion on a plain. This system is equivalent to a single particle of a reduced mass particle of a reduced mass µ, µ, moving around the center of moving around the center of mass, with a constant radius.mass, with a constant radius.

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The Quantum Mechanics Postulates for a Free The Quantum Mechanics Postulates for a Free Particle on a RingParticle on a Ring

1. (The tools of the game) The system state can be described by a 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 wavepacket, (the board of the game) pertaining to the space of continuous functions in angle continuous functions in angle ::

2. (The rules of the game) For each component in the wavepacket 2. (The rules of the game) For each component in the wavepacket the following is true:the following is true:

3. (The interface) the measurement outcome has the following 3. (The interface) the measurement outcome has the following probability of finding the particle in an angle probability of finding the particle in an angle : :

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m=1m=1 m=1.5m=1.5 m=2m=2

Postulates I: Quantization of mPostulates I: Quantization of m

To meet the continuity condition it is possible to To meet the continuity condition it is possible to include in the wavepacket only functions whose include in the wavepacket only functions whose quantum number m is an integer quantum number m is an integer

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Postulates II: Quantization of the Angular Postulates II: Quantization of the Angular Momentum and the EnergyMomentum and the Energy

The dispersion ratio is selected to ccorrospond to the The dispersion ratio is selected to ccorrospond to the classical limit of h:classical limit of h:

To each waveTo each wave are attributedare attributed

the angular momentumthe angular momentum

and the kinetic energyand the kinetic energy

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Postulates III : Dipole MomentPostulates III : Dipole Moment

In a single wave the charge distribution is symmetrical. In In a single wave the charge distribution is symmetrical. In superposition of two waves it is possible to obtain an superposition of two waves it is possible to obtain an asymmetrical distribution of the charge, which is equivalent to asymmetrical distribution of the charge, which is equivalent to an existence of a Dipole Moment an existence of a Dipole Moment

++ ==

yy yy yy

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Quantization of the Light AbsorptionQuantization of the Light Absorption

A rotating Dipole Moment is capable of exchanging energy with A rotating Dipole Moment is capable of exchanging energy with a radiation field in its self frequency (the resonance principle.)a radiation field in its self frequency (the resonance principle.)

The frequency of an envelope changing in a superposition is: The frequency of an envelope changing in a superposition is:

And therefore: And therefore: